# REFS

Every named reference, sequence id, theorem and attribution on this tree's root pages, resolved to a canonical URL. Ids that appear only inside a lab study are resolved in that study's own pages.

- A row is carried here only when it resolves to a URL, a DOI or an OEIS id. Nothing softer is listed.
- Every OEIS id below is checked on both its name and its first terms. None is misattributed.
- One id needs its provenance said out loud. A299916 is defined arithmetically and its Menger geometry lives only in a contributor comment by Albert Safstrom; Wikipedia sources the same claim to that comment and to a newspaper piece, so no peer-reviewed source states it. The geometry is nonetheless true - it is reproved cold from the Menger digit rule, see [README](README.md) - but it is cited as a comment, never as the sequence's definition.
- A000351 and A000420 are the level fills of the void and of corner-and-centre; A003463 and A125833 are Verified rows of the ledger. Every row here is consumed on a page of this tree.

## OEIS SEQUENCES

| ref | title | url |
|---|---|---|
| A000070 | Partition sums: a(n) = Sum_{k=0..n} p(k); the qualifying-signature count of the divisor-avatar material, now the `divisor-avatars` lane | https://oeis.org/A000070 |
| A000290 | The squares, n^2; the odd-side fills of the low-corner design in the plane | https://oeis.org/A000290 |
| A000384 | Hexagonal numbers, n(2n-1); the odd-side fills of the flat tree | https://oeis.org/A000384 |
| A000567 | Octagonal numbers, n(3n-2); the odd-side fills of the flat carpet | https://oeis.org/A000567 |
| A001844 | Centered square numbers, 2n(n+1)+1; the odd-side fills of the flat void | https://oeis.org/A001844 |
| A016754 | Odd squares, (2n+1)^2; the odd-side fills of the solid square, also the centered octagonal numbers | https://oeis.org/A016754 |
| A001481 | Numbers that are the sum of 2 squares; the ring radii squared of a spun square-lattice picture | https://oeis.org/A001481 |
| A003136 | Loeschian numbers, the norms x^2 + xy + y^2; the ring radii squared of a spun hexagonal picture | https://oeis.org/A003136 |
| A004016 | Theta series of the planar hexagonal lattice; the weight of each hexagonal ring | https://oeis.org/A004016 |
| A064533 | Decimal expansion of the Landau-Ramanujan constant K = 0.7642236..., the constant in `#{k <= X : k = a^2 + b^2} ~ K X / sqrt(ln X)` | https://oeis.org/A064533 |
| A004018 | Theta series of the square lattice, r2(n); the weight of each square ring | https://oeis.org/A004018 |
| A000351 | Powers of 5: a(n) = 5^n | https://oeis.org/A000351 |
| A000370 | Number of NPN-equivalence classes of Boolean functions of n or fewer variables | https://oeis.org/A000370 |
| A000420 | Powers of 7: a(n) = 7^n | https://oeis.org/A000420 |
| A000578 | The cubes: a(n) = n^3 | https://oeis.org/A000578 |
| A000616 | a(-1)=1 by convention; for n >= 0, a(n) = number of irreducible Boolean functions of n variables | https://oeis.org/A000616 |
| A001024 | Powers of 15: a(n) = 15^n | https://oeis.org/A001024 |
| A001316 | Gould's sequence: number of odd entries in row n of Pascal's triangle; a(n) = 2^A000120(n) | https://oeis.org/A001316 |
| A002407 | Cuban primes: primes which are the difference of two consecutive cubes | https://oeis.org/A002407 |
| A003180 | Number of equivalence classes of Boolean functions of n variables under action of symmetric group | https://oeis.org/A003180 |
| A003215 | Hex (or centered hexagonal) numbers: 3*n*(n+1)+1; also the lattice lines down the space diagonal of a cube of side n+1, hence the sponge's diagonal shadow at n = 3^level - 1 | https://oeis.org/A003215 |
| A220978 | a(n) = 3^(2n+1) - 3^(n+1) + 1, the left Aurifeuillian factor of 3^(6n+3) + 1; the same diagonal shadow indexed by the level | https://oeis.org/A220978 |
| A003463 | a(n) = (5^n - 1)/4 | https://oeis.org/A003463 |
| A004662 | Powers of 3 written in base 8 | https://oeis.org/A004662 |
| A005418 | Number of (n-1)-bead black-white reversible strings; row sums of Losanitsch's triangle | https://oeis.org/A005418 |
| A005898 | Centered cube numbers: n^3 + (n+1)^3 | https://oeis.org/A005898 |
| A009964 | Powers of 20 | https://oeis.org/A009964 |
| A009971 | Powers of 27 | https://oeis.org/A009971 |
| A011934 | a(n) = abs(1^3 - 2^3 + 3^3 - ... + (-1)^(n+1)*n^3) | https://oeis.org/A011934 |
| A016185 | a(n) = 9^n - 8^n | https://oeis.org/A016185 |
| A016755 | Odd cubes: a(n) = (2*n + 1)^3 | https://oeis.org/A016755 |
| A018413 | Divisors of 363 | https://oeis.org/A018413 |
| A034474 | a(n) = 5^n + 1 | https://oeis.org/A034474 |
| A043635 | Numbers whose base-9 representation has exactly 6 runs; its 30 listed terms lie wholly inside the integer census's miss set | https://oeis.org/A043635 |
| A047999 | Sierpinski's triangle (or gasket): Pascal's triangle mod 2 | https://oeis.org/A047999 |
| A048883 | a(n) = 3^wt(n); number of odd values in the n-th layer of Pascal's tetrahedron | https://oeis.org/A048883 |
| A049537 | Values of k for which A075059(k) = A003418(k) + 1 is prime; the first record to carry a 4-term window of the integer census's miss set, at offset 417 | https://oeis.org/A049537 |
| A054247 | Number of n X n binary matrices under action of the dihedral group D_4 | https://oeis.org/A054247 |
| A065473 | Decimal expansion of the strongly carefree constant, Product_p (1 - (3p-2)/p^3) | https://oeis.org/A065473 |
| A069403 | a(n) = 2*Fibonacci(2*n+1) - 1 | https://oeis.org/A069403 |
| A084237 | Mertens's function M(10^n); the base-10 full-set control column of the Mobius meter census | https://oeis.org/A084237 |
| A100290 | Numbers divisible by the smallest number with the same binary weight, so A038573(a(n)) divides a(n); shares a 12-term window with the integer census's champion set, then gives 21 where the census gives 20 | https://oeis.org/A100290 |
| A103532 | Number of divisors of 240^n; the odd bisection of A011934 | https://oeis.org/A103532 |
| A112820 | Numbers k with lcm(1,2,...,k)/17 equal to the denominator of the k-th harmonic number; carries 20 consecutive integers of the census's miss set | https://oeis.org/A112820 |
| A118471 | a(0)=1, a(n) = a(n-1)*(n+1) if n is in the sequence and a(n-1)+1 otherwise; carries 20 consecutive integers of the census's miss set | https://oeis.org/A118471 |
| A125833 | Numbers whose base-5 representation is 333...3 | https://oeis.org/A125833 |
| A128625 | Expansion of (1+3*x)/(1-5*x) | https://oeis.org/A128625 |
| A129824 | a(n) = Product_{k=0..n} (1 + binomial(n,k)) | https://oeis.org/A129824 |
| A141148 | Number of aperiodic ternary necklaces with n beads of each color and no adjacent beads the same | https://oeis.org/A141148 |
| A154105 | a(n) = 12*n^2 + 18*n + 7 | https://oeis.org/A154105 |
| A192908 | Constant term in the reduction by (x^2 -> x + 1) of a polynomial family; a(n) = 2*Fibonacci(2n-2) + 1 | https://oeis.org/A192908 |
| A209631 | Square array of the exponential transform applied n times to the identity function; the only record carrying the census's written-per-decade run `9, 90, 859`, and it continues 6689 where the census gives 5452 | https://oeis.org/A209631 |
| A229896 | Sizes of logical groups of the same integer in A229895; carries `1, 17, 217, 2465, ...` as an interior window | https://oeis.org/A229896 |
| A255016 | Number of toroidal n X n binary arrays under rotation and/or reflection of rows and/or columns and transposition | https://oeis.org/A255016 |
| A268240 | Pascal's tetrahedron of trinomial coefficients (A046816) read mod 2 | https://oeis.org/A268240 |
| A299916 | Name is `a(n) = A299914(2n+1)` only, offset 0, from Sahin and Tan's divisibility paper; the six-pointed-star hole count is a comment, not the definition. Holes of the `n`-th size, not tiles: this tree's slice census is `A299916(n+1)` | https://oeis.org/A299916 |
| A332705 | Number of unit square faces (surface area) of a stage-n Menger sponge | https://oeis.org/A332705 |
| A336231 | Integers whose binary expansion has an even number of 0's between any two consecutive 1's; shares a 12-term window with the census's champion set, then gives 19 where the census gives 20 | https://oeis.org/A336231 |
| A347825 | Number of ways to cut a 2 X n rectangle into integer-sided rectangles up to symmetry | https://oeis.org/A347825 |
| A361796 | Primes preceded by two consecutive products of four distinct primes; the longest record lying wholly inside the census's miss set, 41 terms | https://oeis.org/A361796 |
| A361870 | Array: nonequivalent 2-colorings of the cells of an n-dimensional hypercube with edges k cells long | https://oeis.org/A361870 |
| A381517 | Perimeter of the Sierpinski carpet at iteration n | https://oeis.org/A381517 |
| A395134 | Decimal expansion of a half-disk chord probability; equals 1 - 16/(3*Pi^2), from Zerr 1891 | https://oeis.org/A395134 |
| A395241 | a(n) = n^2*(4*n + 3) - this tree's own submission, void subcubes of the odd sponge tile | https://oeis.org/A395241 |
| A396922 | E.g.f. A(x) satisfies A(x / A(log(A(log(A(log(A(x)))))))) = exp(x) | https://oeis.org/A396922 |
| A396934 | Number of pairs (i,j) with 0 <= i,j < 2^n, i AND j = 0, and gcd(i,j) = 1 - this tree's own submission | https://oeis.org/A396934 |
| A398348 | Number of toroidal n X n X n binary arrays under per-axis rotation/reflection and axis permutation - this tree's own submission, the base line at dim 3 | https://oeis.org/A398348 |
| A001045 | Jacobsthal sequence, a(n) = a(n-1) + 2*a(n-2); the seed of the run-length transform that counts ON cells of rule 150 | https://oeis.org/A001045 |
| A071053 | Number of ON cells at generation n of elementary rule 150 started from a single ON cell | https://oeis.org/A071053 |
| A087206 | a(n) = 2*a(n-1) + 4*a(n-2); with a(0)=1, a(1)=4; equals 2^n Fibonacci(n+2), the ON cells of rule 150 over its first 2^n rows | https://oeis.org/A087206 |
| A160239 | Number of ON cells at generation n of Fredkin's Replicator, the odd-rule automaton on the eight-cell Moore neighbourhood | https://oeis.org/A160239 |
| A246035 | Number of odd terms in f^n where f = (1/x+1+x)*(1/y+1+y), the nine-cell Moore odd-rule automaton; equals A071053(n)^2, so a separable planar kernel is the tensor square of rule 150 | https://oeis.org/A246035 |
| A000029 | Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets); offset 0, first terms `1, 2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 126` - the record [sequences](sequences.md) reads against `mrlymath::bang::baseq::bracelets`, the base line at dim 1 | https://oeis.org/A000029 |
| A000244 | Powers of 3: a(n) = 3^n; offset 0, first terms `1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147` - the `3^level` record of [sequences](sequences.md), every admissible cut of `bang dim 3, code 126` at base 2, the design credited on [DISCOVERIES](/research/discoveries/) | https://oeis.org/A000244 |
| A000930 | Narayana's cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3); offset 0, first terms `1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41` - the exact ray mass `M_n(1, 12) = A000930(n) - 1` on the supergolden ray of [coprime](notes/coprime.md) | https://oeis.org/A000930 |
| A001018 | Powers of 8: a(n) = 8^n; offset 0, first terms `1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592` - the carpet's level fill `8^level`, key `sequence_dim=2_code=7_measure=fills_axis=level` on [sequences](sequences.md) | https://oeis.org/A001018 |
| A005728 | Number of fractions in Farey series of order n; offset 0, first terms `1, 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43` - equals `1 + sum_(k <= n) phi(k)`, the lit nodes of the [farey](notes/farey.md) stack counted with `0/1`, one more than that page's `m = sum_(k <= Q) phi(k)` on `(0, 1]` | https://oeis.org/A005728 |
| A018805 | Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y)=1}; offset 1, first terms `1, 3, 7, 11, 19, 23, 35, 43, 55, 63, 83, 91` - the totient-sieve count `2*sum_(k = 1..n) phi(k) - 1` of lit points in [pi](notes/pi.md), Verified there to `N = 10000` | https://oeis.org/A018805 |
| A034851 | Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n; offset 0, read by rows with first terms `1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2` - named inside A005418's own name, which [core](notes/core.md) quotes from the live entry for the reversible-string base count | https://oeis.org/A034851 |
| A056040 | Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)); offset 0, first terms `1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772` - the level-one slice row `dim!/floor(dim/2)!^2` from dim 2, an existing entry met from a new direction and nothing to submit, on [README](README.md) and [sequences](sequences.md) | https://oeis.org/A056040 |
| A001037 | Number of degree-n irreducible polynomials over GF(2); number of n-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive period n; number of binary Lyndon words of length n - the Lyndon count of the design's word product | https://oeis.org/A001037 |
| A019554 | Smallest number whose square is divisible by n; multiplicative with `a(p^e) = p^ceiling(e/2)`, Dirichlet series `zeta(2s-1) zeta(s-1)/zeta(2s-2)` on the record; the period of the quadratic lean of the [stack](notes/stack.md) is `lcm(b, a(d))` | https://oeis.org/A019554 |
| A000188 | Number of solutions to `x^2 = 0 mod n`, also the square root of the largest square dividing `n`; first terms `1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2`; the quadratic lean on the [stack](notes/stack.md) has `d* = d/a(d)` for the least `k` with `d` dividing `k^2` | https://oeis.org/A000188 |

## PAPERS AND BOOKS

| ref | title | url |
|---|---|---|
| Huang 2019 | Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture, Annals of Mathematics 190(3) | https://doi.org/10.4007/annals.2019.190.3.6 |
| Glass 1969 | Moire effect from random dots, Nature 223, 578-580 | https://doi.org/10.1038/223578a0 |
| Hardy 1915 | On the expression of a number as the sum of two squares, Quarterly Journal of Mathematics 46, 263-283; the Bessel series for the circle-problem error | https://zbmath.org/?q=an:45.1253.01 |
| Cherny, Anitas, Osipov and Kuklin 2011 | Deterministic fractals: extracting additional information from small-angle scattering data, Phys. Rev. E 84, 036203 | https://doi.org/10.1103/PhysRevE.84.036203 |
| Mattila 1987 | Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34, 207-228 | https://doi.org/10.1112/S0025579300013462 |
| Falconer, Fraser and Jin 2015 | Projections of self-similar and related fractals: a survey of recent developments, Fractal Geometry and Stochastics V | https://doi.org/10.1007/978-3-319-18660-3_4 |
| Huang 2019 preprint | same paper, arXiv:1907.00847 | https://arxiv.org/abs/1907.00847 |
| Sahin and Tan 2018 | Conditional (Strong) Divisibility Sequences, Fibonacci Quarterly 56(1), 18-31 - the source A299914 and A299916 are drawn from; it is number theory and says nothing about Menger sponges | https://www.fq.math.ca/56-1.html |
| Ethier and Lee 2015 | Counting Toroidal Binary Arrays, II, J. Integer Seq. 18, Article 15.8.3 | https://cs.uwaterloo.ca/journals/JIS/VOL18/Lee/lee6.html |
| Nakajima and Watanabe 2026 | Topology of slices through the Sierpinski tetrahedron, arXiv:2603.06004 | https://arxiv.org/abs/2603.06004 |
| Nakajima and Watanabe 2026 journal | same paper, Chaos, Solitons & Fractals 209, 118353 | https://doi.org/10.1016/j.chaos.2026.118353 |
| Marstrand 1954 | Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions, Proc. LMS s3-4(1), 257-302 - almost-every-line slice dimension in the plane, cited on [spectra](notes/spectra.md) | https://doi.org/10.1112/plms/s3-4.1.257 |
| Mattila 1975 | Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 227-244 - the hyperplane version of the same statement, which is the one [spectra](notes/spectra.md) needs in `R^3` | https://doi.org/10.5186/aasfm.1975.0110 |
| Gatzouras 2000 | Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc. 352(5), 1953-1983 | https://doi.org/10.1090/S0002-9947-99-02539-8 |
| Falconer 1995 | On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. 123(4), 1115-1124 | https://doi.org/10.1090/S0002-9939-1995-1224615-4 |
| Kombrink and Winter 2020 | Lattice self-similar sets on the real line are not Minkowski measurable, Ergodic Theory Dynam. Systems 40(1), 221-232 | https://doi.org/10.1017/etds.2018.26 |
| Kombrink and Winter preprint | same paper, arXiv:1801.08595 | https://arxiv.org/abs/1801.08595 |
| Lapidus and van Frankenhuijsen 2006 | Fractal Geometry, Complex Dimensions and Zeta Functions, Springer Monographs in Mathematics | https://doi.org/10.1007/978-0-387-35208-4 |
| Lapidus and Maier 1995 | The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. 52(1), 15-34 - the `(ISP)` equivalence, cited by [dimensions](notes/dimensions.md) | https://doi.org/10.1112/jlms/52.1.15 |
| Barlow and Bass 1999 | Brownian Motion and Harmonic Analysis on Sierpinski Carpets, Canad. J. Math. 51(4), 673-744 - the rigorous foundation under [walks](notes/walks.md)'s carpet row | https://doi.org/10.4153/CJM-1999-031-4 |
| Chow, Varju and Yu 2024 | Counting rationals and Diophantine approximation in missing-digit Cantor sets, arXiv:2402.18395; cited for AD-regularity of missing-digit measures, the box bound behind [coprime](notes/coprime.md)'s dimension-above-one theorem, and for Remark 6.1, the base-3 missing-digit measure's Fourier `l1`-dimension below 1/2, which is what kills the componentwise route to Lemma B; the only power saving of the right shape, its hypotheses covering base `b >= 5` on `b - 1` digits and base 4, excluding base 3 on two digits; Read at source in arXiv:2402.18395v2, 29 pages: Definition (p.6) `hat kappa_t(nu) = sup{s : sum_(xi <= Q) abs(hat nu(xi))^t << Q^(1-s)}`; Theorem 2.1 (p.6) the `hat kappa_1 > 1/2` criterion; Proposition 2.4 (p.7) `hat kappa_1 > 1/2` for `b >= 5` with `b - 1` digits and for `b = 4` with `{0,1,2}` or `{1,2,3}`; Theorem 4.2 (p.11) the sandwich `-log(max_x b^(-L) sum_i S_L(x + i/b^L))/log b^L <= hat kappa_1 <= -log(min_x ...)/log b^L` with both sides converging; Section 6 (p.25) the SageMath interval-arithmetic certificates at `L = 2`, `delta = 10^(-5)`, `tau = 1/2` for `(4,0), (5,0), (5,2), (6,0), (6,1), (6,2)` and `L = 4` for `(5,1)`; Remark 6.1 (pp.25-26) verbatim: `We note that hat kappa_1(nu) < 1/2 for the remaining choices of parameters, namely (b,a) in {(3,0),(3,1),(3,2),(4,1),(4,2)}`, shown by (6.5) `min_x F_L(x) > b^((1-tau)L) + delta 2 b^L (b^L - 1) pi` with `L = 2`, `delta = 10^(-4)`, `tau = 1/2`. So every base-3 two-digit measure has `l^1` dimension below `1/2` and the base-4 set `{0,1,2}` above it; the ledger's Conjecture row on the componentwise route is Verified at source | https://arxiv.org/abs/2402.18395 |
| Erdos, Mauduit and Sarkozy 1998 | On arithmetic properties of integers with missing digits I: distribution in residue classes, J. Number Theory 70, 99-120, doi:10.1006/jnth.1998.2229 | https://www.semanticscholar.org/paper/On-Arithmetic-Properties-of-Integers-with-Missing-Erdos-Mauduit/819d346a221f620ec9107933f0acc22cd345928d |
| Konyagin 2001 | Arithmetic properties of integers with missing digits: distribution in residue classes, Period. Math. Hungar. 42, 145-162, doi:10.1023/A:1015256809636 | https://link.springer.com/article/10.1023/A:1015256809636 |
| Maynard 2019 | Primes with restricted digits, Invent. Math. 217, 127-218, doi:10.1007/s00222-019-00865-6 - the reference point for primes in the gasket. Read at source in arXiv:1604.01041v2: Theorems 1.1-1.2 (pp.2-3; base 10 with one excluded digit, and base `q` large with `s <= q^(23/80)` excluded digits, `s <= q - q^(57/80)` when consecutive); Proposition 7.1 (p.18), the Type I estimate for the SET: `sum_(q < Q, (q,10)=1) abs(#{a in A : q divides a, (a,10) = 1} - kappa #A/q) <<_A #A (log X)^(-A)` for `Q <= X^(50/77) (log X)^(-2A-2)`, residue `0`, all moduli coprime to 10, a log saving, and the proof (pp.26-27) is residue-uniform; Lemma 8.1 (p.25) the large sieve `sum_(q <= Q) sum_((a,q)=1) F_Y(a/q) << Q^(54/77) + Q^2 Y^(-50/77)`; Lemma 8.2 (p.26) the `l^infinity` bound `exp(-c log Y/log q)` for `q < Y^(1/3)` with a factor coprime to 10; Lemma 10.3 (pp.36-37) the `l^1` bounds `int F_Y << Y^(-50/77)` and `sup_beta sum_(a < Y1) F_(Y2)(beta + a/Y3) << Y1^(27/77)`; Section 16 (pp.68-69) the general-base `l^1` bound `((q log q + q s)/(q - s))^k`, which is `O(Y^(23/80+eps))` for `s <= q^(23/80)`, and `((q log q + q - s)/(q - s))^k` for consecutive excluded digits, `Y^(23/80+eps)` at `s <= q - q^(57/80)`; that section is the author's own sketch, `leaving the complete details to the interested reader`, and it states no Type I proposition for general `q`, only `We can use this bound in place of Lemma 10.3 and Lemma 10.4 throughout the argument with the same (or stronger) consequences`. The level `50/77` is one minus the `l^1` exponent `27/77`, and that exponent is the Markov eigenvalue bound (10.5), `lambda_(1,4) < 2.24190 < 10^(27/77)` for every choice of excluded digit. A whole-text search of the 70-page source finds `Mobius` once, in `By Mobius inversion` opening the proof of Proposition 7.1 on p.26, and finds `Mertens` and `Liouville` nowhere; Sieve sections now read at source in arXiv:1604.01041v2: Section 2 (pp.3-5) the outline, the `l^1` bound (2.1) `sum_(a<X) abs(S_A(a/X)) << #A X^0.36`, the Type II range `N in [X^0.36, X^0.425]` too narrow for an asymptotic, the Harman minorant, and the remark that GRH gives only `S_P << X^(3/4+o(1))` pointwise; Section 6 (pp.8-17) the decomposition with cut points `z_1..z_6`, Buchstab to four primes, nine integrals `I_1..I_9` bounded `0.02895, 0.35718, 0.01402, 0.04238, 0.05547, 0.06622, 0.21879, 0.20339, 0.00924`, sum `< 0.996`, giving (6.17) `#{p in A} >= (1+o(1)) kappa_A #A/(1000 log X)` with a Mathematica file on arXiv; Section 7 (pp.17-25) Proposition 7.2 the Type II estimate for polytope-counted `1_R` with a coordinate subset summing into `[9/25 + eps, 17/40 - eps]` or by symmetry `[23/40 + eps, 16/25 - eps]`, Lemma 7.4 the fundamental lemma at level `X^(50/77 - eps)` applied for `d <= X^(1 - theta_1)`; Section 8 (pp.25-27) the proof of 7.1 by Mobius inversion over the divisors `d` | https://link.springer.com/article/10.1007/s00222-019-00865-6 |
| Banks and Shparlinski 2004 | Arithmetic properties of numbers with restricted digits, Acta Arithmetica, doi:10.4064/aa112-4-1, whose identifier names 112(4), 313-332 - coprime pairs in digit-restricted sets, the one-dimensional ellipsephic ancestor of [coprime](notes/coprime.md); the volume is elsewhere given as 113(4), 313-328 and neither reading is checked at the article | https://doi.org/10.4064/aa112-4-1 |
| Zucker 1974 | Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A 7(13), 1568-1575 | https://doi.org/10.1088/0305-4470/7/13/011 |
| Borwein et al. 2013 | Lattice Sums Then and Now, Encyclopedia of Mathematics and its Applications 150, Cambridge | https://doi.org/10.1017/CBO9781139626804 |
| Franel 1924 | Les suites de Farey et le probleme des nombres premiers, Gott. Nachr., 198-201 | https://eudml.org/doc/59156 |
| Landau 1924 | Bemerkungen zu der obenstehenden Abhandlung von J. Franel, Gott. Nachr., 202-206 | https://eudml.org/doc/59157 |
| Edwards 1974 | Riemann's Zeta Function, Academic Press; the Farey material is chapter 12, section 12.2 | https://archive.org/details/riemannszetafunc00edwa_0 |
| Glaisher 1899 | On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quart. J. Pure Appl. Math. 30, 150-156 | https://babel.hathitrust.org/cgi/pt?id=hvd.32044102924578 |
| Alaoglu and Erdos 1944 | On highly composite and similar numbers, Trans. Amer. Math. Soc. 56, 448-469, doi:10.1090/S0002-9947-1944-0011087-2; the source of the colossally abundant construction behind the Robin-frontier ladders | https://doi.org/10.1090/S0002-9947-1944-0011087-2 |
| Robin 1984 | Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. 63, 187-213; the inequality `sigma(n) < e^gamma n log log n` for `n > 5040` that the Robin corridor is named for. No DOI and no arXiv copy; the zbMath record is the permalink | https://zbmath.org/0516.10036 |
| Baez-Duarte 2005 | A sequential Riesz-like criterion for the Riemann hypothesis, Int. J. Math. Math. Sci., 3527-3537; the source of the coefficient `c_k = sum_n mu(n) n^{-2} (1 - n^{-2})^k`, and one of the closed RH routes, where the stack-brightness meter is illustration only | https://doi.org/10.1155/IJMMS.2005.3527 |
| Rodgers and Tao 2020 | The de Bruijn-Newman constant is non-negative, Forum of Mathematics Pi 8, e6 - another closed RH route | https://doi.org/10.1017/fmp.2020.6 |
| Mullner 2017 | Automatic sequences fulfill the Sarnak conjecture, Duke Math. J. 166(17), 3219-3290 - the theorem that makes the digit-restricted Mobius question well-posed | https://doi.org/10.1215/00127094-2017-0024 |
| Mauduit and Rivat 2010 | Sur un probleme de Gelfond: la somme des chiffres des nombres premiers, Ann. of Math. 171(3), 1591-1646 - sum-of-digits against the primes; no page of this tree consumes it now | https://doi.org/10.4007/annals.2010.171.1591 |
| Hochman and Shmerkin 2012 | Local entropy averages and projections of fractal measures, Ann. of Math. 175(3), 1001-1059 - the weights-instead-of-0-1 rung of the generalization ladder | https://doi.org/10.4007/annals.2012.175.3.1 |
| Shmerkin 2019 | On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions, Ann. of Math. 189(2), 319-391 - one of the two independent proofs of the Furstenberg slice conjecture cited on [crop](notes/crop.md), never claimed | https://arxiv.org/abs/1609.07802 |
| Wu 2019 | A proof of Furstenberg's conjecture on the intersections of xp- and xq-invariant sets, Ann. of Math. 189 (2019); the arXiv page states the Annals acceptance without volume or pages, so no page numbers are quoted - the other independent proof of the same conjecture, cited on [crop](notes/crop.md) | https://arxiv.org/abs/1609.08053 |
| Turan 1949 | On a new method in the analysis with applications, Cas. Pest. Mat. Fys. 74, 123-126; cited in [coprime](notes/coprime.md) only to record that power sums lower-bound extremal eigenvalues and cannot supply the ray machine's upper bound | https://doi.org/10.21136/CPMF.1949.133455 |
| Montgomery and Vaughan 1973 | The large sieve, Mathematika 20, 119-134; the analytic model named for the octave deficit-count encoding behind Statement (A), now the `gasket-ray-machine` lane | https://doi.org/10.1112/S0025579300004708 |
| Allouche and Shallit 1992 | The ring of k-regular sequences, Theoret. Comput. Sci. 98(2), 163-197; the source of the finite-range theorem that forbids `d(n)` and `sigma(n)` from being k-automatic in any base | https://doi.org/10.1016/0304-3975(92)90001-v |
| Allouche and Shallit 2003 | Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press - the one-letter specialization of the same object | https://doi.org/10.1017/CBO9780511546563 |
| Rigo 2020 | From combinatorial games to shape-symmetric morphisms, Lecture Notes in Mathematics, 227-291; the survey behind the k-regular framing of unbounded census observables | https://doi.org/10.1007/978-3-030-57666-0_5 |
| Berstel and Reutenauer 2011 | Noncommutative Rational Series with Applications, Cambridge University Press - the Hankel-rank criterion used to build the mixed-product representations | https://doi.org/10.1017/CBO9780511760860 |
| Schutzenberger 1961 | On the definition of a family of automata, Information and Control 4(2-3), 245-270 - finite Hankel rank over a free monoid is equivalent to a linear representation | https://doi.org/10.1016/S0019-9958(61)80020-X |
| Jungers 2009 | The Joint Spectral Radius: Theory and Applications, Springer LNCIS 385 - the asymptotic invariant for a noncommuting matrix product along an arbitrary word | https://doi.org/10.1007/978-3-540-95980-9 |
| Furstenberg and Kesten 1960 | Products of random matrices, Annals of Mathematical Statistics 31(2), 457-469 - the Lyapunov exponent for random or ergodic schedules | https://doi.org/10.1214/aoms/1177705909 |
| Krattenthaler 1999 | Advanced Determinant Calculus, Seminaire Lotharingien de Combinatoire B42q; cited for determinants of binomial-coefficient matrices, the shape the threshold determinant `d_dim` takes | https://www.mat.univie.ac.at/~slc/wpapers/s42kratt.html |
| Holte 1997 | Carries, combinatorics, and an amazing matrix, Amer. Math. Monthly 104(2), 138-149 - the classical carries transfer matrix, `P(i,j) = P{jb <= i + X_1 + ... + X_n <= (j+1)b-1}` on states `0..n-1` with spectrum `{1, 1/b, ..., 1/b^(n-1)}` and Eulerian-number stationary vector. The Monthly original is behind JSTOR; it is read here through Diaconis and Fulman's verbatim (H1)-(H6) restatement. THE EARLIER "is an instance" CLAIM IS WITHDRAWN: the slice automaton is a digit-restricted, digit-pinned ANALOGUE, not an instance. Removing both the joint digit restriction and the output-digit pinning recovers `base^dim` times Holte's matrix exactly, same indexing; keeping them gives an irrational Perron root, which Holte's integer spectrum forbids | https://doi.org/10.2307/2974981 |
| Diaconis and Fulman 2009 | Carries, shuffling, and an amazing matrix, Amer. Math. Monthly 116(9), 788-803 - the free account of Holte's (H1)-(H6) and the identity of the carries chain with the riffle-shuffle descent process; the source this tree quotes Holte through | https://arxiv.org/abs/0806.3583 |
| Diaconis and Fulman 2012 | Foulkes characters, Eulerian idempotents, and an amazing matrix, J. Algebraic Combin. 36(3), 425-440 - left eigenvectors are the Foulkes character table, right eigenvectors the Eulerian idempotents | https://arxiv.org/abs/1102.5159 |
| Diaconis and Fulman 2014 | Combinatorics of balanced carries, Adv. Appl. Math. 55, 1-16 - the balanced-digit analogue, spectrum still `1, 1/b, ..., 1/b^n`, hyperoctahedral Foulkes characters and Eulerian idempotents; second witness that changing the digit set within the independent frame preserves Holte's spectrum | https://arxiv.org/abs/1309.5116 |
| Nakano and Sadahiro 2014 | A generalization of carries process and Eulerian numbers, Adv. Appl. Math. 53, 28-43 - the nearest published generalisation of the digit set (shifted consecutive sets `{d,...,d+b-1}`, negative bases); its summand digits are explicitly independent, which is exactly what the mrly restriction breaks | https://arxiv.org/abs/1306.2790 |
| Forster and Nagy 2000 | On nonnegative realizability of partitioned spectra, Linear Algebra Appl. 311 - staged for the spectral separation lemma; no claim on this tree consumes it | https://doi.org/10.1016/S0024-3795(00)00089-6 |
| Seneta 2006 | Non-negative Matrices and Markov Chains, revised printing, Springer - the Perron-Frobenius and Collatz-Wielandt machinery the slice-dimension sections lean on | https://doi.org/10.1007/0-387-32792-4 |
| Garcia-Armas, Ghorpade and Ram 2011 | Relatively prime polynomials and nonsingular Hankel matrices over finite fields, J. Combin. Theory Ser. A 118(3), 819-828 - the exact, non-asymptotic unrestricted coprimality count that gives the `1 - 1/q` baseline; the page named it by description rather than title, so the identification here is inferred | https://doi.org/10.1016/j.jcta.2010.11.005 |
| Weil 1948 | Sur les courbes algebriques et les varietes qui s'en deduisent, Hermann, Paris - the Riemann hypothesis over function fields, the theorem that makes the `F_q[t]` diagnostic decisive | https://mathscinet.ams.org/mathscinet/relay-station?mr=0027151 |
| Rosen 2002 | Number Theory in Function Fields, Springer GTM 210 - the standard reference for the `F_q[t]` zeta function and its Euler product | https://doi.org/10.1007/978-1-4757-6046-0 |
| Mertens 1897 | Ueber eine zahlentheoretische Function, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, 106, 761-830; the function the Mobius-weighted stack on [farey](notes/farey.md) renders at every node | https://www.zobodat.at/pdf/SBAWW_106_2a_0761-0830.pdf |
| Deleglise and Rivat 1996 | Computing the summation of the Mobius function, Experiment. Math. 5(4), 291-295 - the `x^(2/3)` Mertens algorithm behind the weighted stack's open status on [farey](notes/farey.md) | https://doi.org/10.1080/10586458.1996.10504594 |
| Odlyzko and te Riele 1985 | Disproof of the Mertens conjecture, J. reine angew. Math. 357, 138-160 - the exhaustive-verification lesson in [method](notes/method.md) | https://doi.org/10.1515/crll.1985.357.138 |
| Griffin, Ono, Rolen and Zagier 2019 | Jensen polynomials for the Riemann zeta function and other sequences, PNAS 116(23), 11103-11110 - cited by [method](notes/method.md) as a false friend, never as support | https://doi.org/10.1073/pnas.1902572116 |
| Mayer 1991 | The thermodynamic formalism approach to Selberg's zeta function for PSL(2,Z), Bull. Amer. Math. Soc. 25(1), 55-60 - the transfer-operator-to-zeta bridge on the mediant rung, cited by [farey](notes/farey.md) | https://doi.org/10.1090/S0273-0979-1991-16023-4 |
| Lagarias 1985 | The computational complexity of simultaneous Diophantine approximation problems, SIAM J. Comput. 14(1), 196-209 - the variable-dimension NP-completeness on [farey](notes/farey.md)'s addressability section | https://doi.org/10.1137/0214016 |
| Barvinok 1994 | A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res. 19(4), 769-779 - the fixed-dimension P placement on [farey](notes/farey.md) | https://doi.org/10.1287/moor.19.4.769 |
| Garey and Johnson 1979 | Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman - problem SP3, Simultaneous Incongruences, the NP-complete neighbour on [farey](notes/farey.md) | https://dl.acm.org/doi/10.5555/574848 |
| Bach, Miller and Shallit 1986 | Sums of divisors, perfect numbers, and factoring, SIAM J. Comput. 15(4), 1143-1154 - sigma-to-factorization, carried on the complexity-frontier line of [DISCOVERIES](/research/discoveries/) | https://doi.org/10.1137/0215080 |
| Zerr 1891 | Solution to Problem 11134, Mathematical Questions and Solutions from the "Educational Times" 55, 161 - the original half-disk chord problem, cited through A395134's own link | https://oeis.org/A395134 |
| Santalo 2004 | Integral Geometry and Geometric Probability, 2nd ed., Cambridge University Press - the Blaschke-Petkantschin formula and chord-power integrals used by the Zerr decomposition | https://doi.org/10.1017/CBO9780511617331 |
| Baake and Huck 2015 | Ergodic properties of visible lattice points, Proceedings of the Steklov Institute of Mathematics 288, 184-198 - visible lattice points as model sets | https://doi.org/10.1134/S0081543815010113 |
| Goins, Harris, Kubik and Mbirika 2018 | Lattice Point Visibility on Generalized Lines of Sight, American Mathematical Monthly 125(7), 593-601, doi:10.1080/00029890.2018.1465760 - the density of points visible along `y = a*x^b` is `1/zeta(b+1)`, the theorem [coprime](notes/coprime.md)'s b-visibility section reproduces; this tree cites the preprint, and the journal details come from the catalogue record | https://arxiv.org/abs/1712.09155 |
| Lindemann 1882 | Ueber die Zahl Pi, Mathematische Annalen 20, 213-225 - the transcendence of `Pi`, which blocks the mismatch theorem at even `D >= 4` | https://doi.org/10.1007/BF01446522 |
| Apery 1979 | Irrationalite de zeta(2) et zeta(3), Asterisque 61, 11-13 - blocks the mismatch theorem at dim 3 against Version L, and the C-finite corollary at dim 3 | https://eudml.org/doc/94858 |
| Rivoal 2000 | La fonction zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris 331(4), 267-270 - why the mismatch theorem stays conditional at odd `D >= 5` | https://arxiv.org/abs/math/0008051 |
| Zudilin 2001 | One of the numbers zeta(5), zeta(7), zeta(9), zeta(11) is irrational, Russian Math. Surveys 56(4), 774-776 - the same conditionality, sharpened | https://doi.org/10.1070/RM2001v056n04ABEH000427 |
| Kenyon 1997 | Projecting the one-dimensional Sierpinski gasket, Israel Journal of Mathematics 97, 221-238 - the projection dichotomy for the dimension-one gasket, whose dimension-below-one half holds every occupied direction | https://doi.org/10.1007/BF02774038 |
| Athreya, Reznick and Tyson 2019 | Cantor set arithmetic, American Mathematical Monthly 126, 4-17 - the quotient set of the middle-thirds set is a union of intervals, so no fractal-geometric input can supply the saving | https://arxiv.org/abs/1711.08791 |
| Yu 2021 | Rational points near self-similar sets - counts rationals near a self-similar set, not on it, the gap that keeps it from the lemma | https://arxiv.org/abs/2101.05910 |
| Flajolet and Odlyzko 1990 | Random mapping statistics, EUROCRYPT '89, LNCS 434, 329-354, doi:10.1007/3-540-46885-4_34 - cited as a MODEL only, for the `Theta(sqrt N)` rho length and reachable-set size of a random mapping on `N` nodes, which is the shape the band automaton's mean reach follows in `lab/py/ratio-set-saving`; no theorem is imported, the band automaton being deterministic and arithmetic rather than random | https://doi.org/10.1007/3-540-46885-4_34 |
| Schleischitz 2021 | On intrinsic and extrinsic rational approximation to Cantor sets, Ergodic Theory and Dynamical Systems 41, 1560-1589 - intrinsic approximation on missing-digit sets, the nearest framing of rationals lying on the set | https://arxiv.org/abs/1812.10689 |
| Moran 1946 | Additive functions of intervals and Hausdorff measure, Math. Proc. Cambridge Philos. Soc. 42(1), 15-23 - Theorem II: if `E = union E_i` with the `E_i` closed, non-overlapping and similar to `E` in ratios `t_i`, then `dim_H E = p_0` with `sum t_i^p_0 = 1` and `0 < H^p_0(E) < infinity`. THIS IS NOT THE VARYING-RATIO CONSTRUCTION: Theorem III reuses the same ratio list `t_1, ..., t_n` at every level and lets only the placement vary, the level-`2` maps not needing to be compositions of the level-`1` maps. Level-varying ratios are the later Moran-set literature, which carried Moran's name for that case long before Rempe-Gillen and Urbanski, who only record the usage rather than coin it - their text says that when all the maps are affine similarities the system `is also referred to as a Moran set construction`, their bibliography sends it to Wen, Moran sets and Moran classes, Chinese Sci. Bull. 46(22), 1849-1856 (2001), and they describe that article as a survey of results known at the time; Feng, Wen and Wu had already titled the objects homogeneous Moran sets in 1997. Cambridge Core serves bibliography and a first-page extract only; the theorem statements are read through the verbatim restatement in Fernandez-Martinez, Guirao and Rodriguez-Bermudez, which follows Moran's own notation | https://doi.org/10.1017/S0305004100022684 |
| Feng, Wen and Wu 1997 | Some dimensional results for homogeneous Moran sets, Science in China Ser. A 40(5), 475-482 - the homogeneous Moran class `M(I_0, {n_k}, {c_k})` is the level-varying construction on the line with `n_k` pieces and one common ratio `c_k` at level `k`, which is a word's shape. Theorem 2.1 gives the Hausdorff dimension of an individual set outright: writing the class in the digit notation of the citing papers, `N_j` subdivisions of which `K_j` are kept, the evenly gapped homogeneous member has `dim_H = s_1 = liminf_j log(K_1...K_j) / log(N_1...N_j)` and the packed-to-one-end partial homogeneous member has `dim_H = s_2 = liminf_j log(K_1...K_j) / (log(N_1...N_j) + log(N_{j+1}/K_{j+1}))`, and Lemma 2.2 bounds every other member by `s_2 <= dim_H E <= s_1`. So both an individual-set formula and the range over the class are proved, and the clean ratio `sum log n_i / sum log(1/c_i)`, read as a liminf, is the Hausdorff dimension of the homogeneous member sitting at the TOP of that range, not a box-dimension expression above it. Theorem 2.1 and Lemma 2.2 are carried from the verbatim restatement in Lai, Perfect fractal sets with zero Fourier dimension and arbitrarily long arithmetic progressions, arXiv:1606.06684. The article is unreachable - Springer is paywalled and the publisher's own PDF mirror is over its bandwidth limit - so the volume, issue and pages are checked at Crossref and in two independent reference lists, and the content is catalogued from restatements | https://doi.org/10.1007/BF02896955 |
| Mauldin and Williams 1988 | Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309(2), 811-829 - construction matrix `A = [t_{i,j}]` from the edge ratios, `A_beta = [t_{i,j}^beta]`, and `Phi(beta)` its spectral radius. Theorem 3: for `G` strongly connected the construction object has `dim_H K = alpha` where `Phi(alpha) = 1`, with `0 < H^alpha(K) < infinity`. Theorem 4: in general `alpha = max{alpha_H}` over the strongly connected components, `H^alpha(K)` is sigma-finite, and is finite exactly when the components attaining `alpha` are pairwise incomparable. A cycle is defined there and is always a strongly connected component, which is the graph-directed reading of a periodic word; the collapse to one composite tile is a consequence of Theorem 3 and is not stated in the paper, and the setup carries one similarity per ordered vertex pair, so several children per level need duplicated vertices | https://doi.org/10.2307/2000940 |
| Rempe-Gillen and Urbanski 2016 | Non-autonomous conformal iterated function systems and Moran-set constructions, Trans. Amer. Math. Soc. 368(3), 1979-2017, doi:10.1090/tran/6490 - the general umbrella for level-varying rules, where the contractions applied at each step in time are allowed to vary, and the source of the naming: the paper itself says that when all the maps are affine similarities the system is also called a Moran set construction. Definition 5 sets the lower and upper pressure as the `liminf` and the `limsup` of `(1/n) log Z_n` separately, Definition 7 takes the Bowen parameter `B` from the lower one, and Theorem 1.1 gives Bowen's formula `HD(J(Phi)) = B(Phi)` under sub-exponential growth of the alphabet | https://arxiv.org/abs/1210.7469 |
| Cristea and Steinsky 2010 | Connected generalised Sierpinski carpets, Topology and its Applications 157(7), 1157-1162 - a generalised Sierpinski carpet is a plane set defined by a sequence of patterns, an `m_k x m_k` pattern at step `k`, with distinct patterns and distinct `m_k` allowed at distinct steps, so the family is level-varying by construction; the paper gives necessary and sufficient conditions for such a carpet to be connected in the Euclidean topology, by graph-theoretical arguments on the cutting types of the patterns. The publisher text is blocked to fetchers, so the definition, the statement and the method are read in the authors' own companion paper on totally disconnected generalised carpets, arXiv:1303.4883 | https://doi.org/10.1016/j.topol.2010.02.005 |
| Cristea and Steinsky 2017 | Mixed labyrinth fractals, Topology and its Applications 229, 112-125 - the family that owns the word mixed: objects constructed from sequences of labyrinth patterns, in general not self-similar, shown to be dendrites, with the paths in the prefractal graphs and the arcs in the fractal studied for path length, box-counting dimension and arc length, and connected to the generalised Sierpinski carpet results. The open copy carrying the same abstract and DOI is arXiv:2009.12206, whose journal-ref misprints the volume as 22 | https://doi.org/10.1016/j.topol.2017.06.022 |
| Barnsley, Hutchinson and Stenflo 2008 | V-variable fractals: fractals with partial self similarity, Advances in Mathematics 218(6), 2051-2088, doi:10.1016/j.aim.2008.04.011 - a code tree is `V`-variable at `V = 1` exactly when all its values at a given level are equal, which is precisely a randomised schedule, and the paper names that case the homogeneous random fractals, with `V -> infinity` giving the standard random fractals. THE DIMENSION THEORY IS NOT IN THIS PAPER: it proves existence, uniqueness and approximation under average contractivity, and Remark 9.2 announces the dimension computation as forthcoming, associating a `V x V` matrix to each state and using Furstenberg-Kesten theory for products of random matrices; the computation is carried out in the companion V-variable fractals: dimension results, Forum Math. 24(3), 445-470 | https://arxiv.org/abs/0802.0064 |
| Barnsley, Hutchinson and Stenflo 2012 | V-variable fractals: dimension results, Forum Mathematicum 24(3), 445-470, doi:10.1515/form.2011.075 - the companion that carries the dimension theory the 2008 paper only announces: it computes the almost sure Hausdorff dimension of `V`-variable fractals satisfying the uniform open set condition, the tools being the notion of a neck, which gives spatial homogeneity at various levels of magnification, and a variant of the Furstenberg-Kesten theorem for products of random `V x V` matrices. The families interpolate between the random homogeneous fractals at `V = 1`, which is the randomised-schedule reading inherited from the 2008 paper, and the random recursive fractals as `V -> infinity`. The De Gruyter text is paywalled, so the abstract and the introduction are read in the authors' own copy at maths-people.anu.edu.au | https://doi.org/10.1515/form.2011.075 |
| Smilansky and Solomon 2021 | Multiscale substitution tilings, Proc. London Math. Soc. 123(6), 517-564 - substitution schemes on a finite set of prototiles in which multiple distinct scaling constants are allowed, several ratios inside one subdivision level, the operation no mrly move can express; an added irrationality assumption on the scaling constants is what makes the resulting tilings and tiling spaces intrinsically different from the standard substitution setup, Penrose and pinwheel included. The open copy is arXiv:2003.11735 | https://doi.org/10.1112/plms.12404 |
| Berthe and Delecroix 2014 | Beyond substitutive dynamical systems: S-adic expansions, RIMS Kokyuroku Bessatsu B46, 81-123 - an S-adic representation writes `u = lim sigma_0 sigma_1 ... sigma_{n-1}(a_n)` and calls `(sigma_n)` the directive sequence, which is what a magic word is. Theorem 2.5: every linearly recurrent symbolic dynamical system is uniquely ergodic. Theorem 3.10: the cone `lim M_0 M_1 ... M_n R_+^d` is the convex hull of the letter-frequency vectors of the words in the system, so frequencies come from the infinite matrix product. Sturmian slopes are parametrised by the Gauss continued-fraction expansion | https://arxiv.org/abs/1309.3960 |
| Fraser 2012 | Inhomogeneous self-similar sets and box dimensions, Studia Mathematica 213(2), 133-156 - the false friend: the inhomogeneous attractor is the compact `F_C = union_i S_i(F_C) union C` for one fixed condensation set `C` unioned in at every step, the homogeneous case being `C` empty, and nothing in it is level-varying. Corollary 2.2 gives `upper-box F_C = max{upper-box F_empty, upper-box C}` under the strong open set condition; Theorems 2.7 and 2.8 bound the lower box dimension and show it does not follow that pattern, behaving far more strangely than the upper box, Hausdorff and packing dimensions | https://arxiv.org/abs/1301.1881 |
| Voet and De Novellis 2025 | Identifying Kronecker product factorizations, arXiv 2510.25292 - exact factorisation of binary matrices under equality, read at source: Lemma 2.4 proves fixed-size uniqueness of a factorisation when it exists (the block reading, with the Van Loan and Pitsianis rearrangement in Remark 3.7 and the scalar `alpha, alpha^-1` caveat over the reals), Definitions 2.2 and 2.3 set up prime matrices and prime decompositions with primality automatic at prime size, Example 2.5 gives three factorisations of one matrix at permuted sizes and Example 2.6 one matrix factoring at sizes `(3,4)` and `(2,2,3)`, and the paper states plainly that a prime decomposition need not have prime sizes; the tree's block lemma and factorisation non-uniqueness are rediscoveries of this and are cited to it | https://arxiv.org/abs/2510.25292 |
| Kempner 1914 | A curious convergent series, Amer. Math. Monthly 21(2), 48-50 - the sum of the reciprocals of the positive integers whose decimal expansion contains no occurrence of a fixed nonzero digit converges. JSTOR and Taylor and Francis both refuse fetchers, so the original is not read here; the statement and the bibliographic record are carried from two independent verbatim restatements, Allouche, Hu and Morin 2024 and Burnol 2026, which print the same volume, issue and pages | https://doi.org/10.2307/2972074 |
| Kohler and Spilker 2009 | Dirichlet-Reihen zu Kempners merkwurdiger konvergenter Reihe, Math. Semesterber. 56(2), 187-199 - Satz 2: for `b >= 2` and a non-empty digit set `D` inside `[0, b-1]` with `D` not equal to `{0}`, the Dirichlet series over the integers whose base-`b` digits all lie in `D` has abscissa of convergence exactly `log(card D) / log b`. This is the earliest source for the abscissa the digit-restricted zeta lives on. The Springer text is paywalled, so Satz 2 is read through its verbatim restatement as Theorem 2 in Allouche, Shallit and Stipulanti 2025 | https://doi.org/10.1007/s00591-009-0059-5 |
| Nathanson 2021 | Dirichlet series of integers with missing digits, J. Number Theory 222, 30-37, doi:10.1016/j.jnt.2020.10.002, read at source in arXiv:2010.06295v2 - the digit rule is allowed to vary with the POSITION, a proper subset `U_i` of `[0, g-1]` being forbidden at place `i`, and the abscissa of convergence of the resulting series is computed; the constant rule `U_i = {u}` gives `sigma_c = log(g-1)/log g` with divergence AT `sigma_c` | https://arxiv.org/abs/2010.06295 |
| Allouche, Mendes France and Peyriere 2000 | Automatic Dirichlet series, J. Number Theory 81(2), 359-373, read at source in the authors' own preprint at webusers.imj-prg.fr. Theorem 3: for `d >= 2` and any `d`-automatic sequence `(u_n)` with values in `C`, the series `sum u_n (n+1)^(-s)` and `sum u_n n^(-s)` are first components of Dirichlet vectors obeying an infinite functional equation got by splitting `n` on its last base-`d` digit, and both continue meromorphically to the whole complex plane with poles, if any, on a finite number of left semi-lattices; the proof places them at `s = log(lambda)/log d + 2 i k pi / log d - l + 1` for `lambda` an eigenvalue of `A = d^(-1)(A_0 + ... + A_(d-1))`, `k` in `Z`, `l` in `N`. This is the theorem a digit-restricted zeta inherits, and its lattice period `2 pi i / log d` is the complex-dimensions period | https://doi.org/10.1006/jnth.1999.2487 |
| Coons 2010 | (Non)automaticity of number theoretic functions, J. Theor. Nombres Bordeaux 22(2), 339-352 - Theorem 3.1 carries the previous conclusion from `k`-automatic to `k`-regular sequences by the same proof. Theorem 2.3, credited to Allouche, runs it backwards: `(mu(n))` is not `k`-automatic for any `k >= 2`, because `1/zeta(s)` has `asymp T log T` poles up to height `T` and those cannot sit on finitely many left semi-lattices. That is why a Mobius-weighted digit-restricted series inherits no continuation from the automatic machinery | https://doi.org/10.5802/jtnb.718 |
| Allouche, Shallit and Stipulanti 2025 | Combinatorics on words and generating Dirichlet series of automatic sequences, Discrete Math. 348(8), 114487, doi:10.1016/j.disc.2025.114487, read at source in arXiv:2401.13524v4 - the unification of the missing-digit Dirichlet series family, restating Kohler and Spilker's abscissa as Theorem 2 and Nathanson's positional version as Theorem 3, and restating the continuation with candidate poles at `z_(n,l)(gamma) = log(gamma)/log b - l + 2 n pi i / log b` over the eigenvalues `gamma` of the sum matrix. It says plainly that proving a candidate is a genuine pole `might turn out to be complicated`, that under primitivity of the sum matrix with non-negative terms only `log(rho)/log b` is known to be a simple pole, and it leaves characterising the other poles as an open problem | https://arxiv.org/abs/2401.13524 |
| Burnol 2026 | On the analytic continuation of Dirichlet series with missing digits, arXiv:2602.19727v2 - the object itself: `K(s) = sum' n^(-s)` over the integers whose base-`b` digits all lie in `A`, with `N = card A` and `A` not equal to `{0}`. Abscissa `s_0 = log_b N`. Proposition 4.1: `K` continues meromorphically to `C`, obeys `(1 - N b^(-s)) K(s) = sum_(a in A minus {0}) a^(-s) + sum_(m >= 1) (-1)^m ((s)_m / m!) b^(-s-m) gamma_m K(s+m)` with `gamma_m = sum_(a in A) a^m`, has a simple pole at `s_0` of positive residue, and has only simple poles, all among `s_(m,k) = s_0 - m + 2 k pi i / log b`, while `prod_(m >= 0) (1 - N b^(-s-m)) K(s)` is entire. Proposition 7.1: a vanishing residue at `s_(0,k)` forces vanishing at every `s_(m,k)`. Theorem 7.4: the exponential generating function of the normalised real-axis residues is the multiplicative INVERSE of the moment generating function of the natural measure on the matching Cantor set, so those residues generalise the Bernoulli numbers. The continuation itself is credited to Allouche, Mendes France and Peyriere. No Mobius or Mertens sum appears anywhere in it. Proposition 5.1: the off-real residue `lambda_(0,k)` equals `(log b)^(-1)` times the limit over `l` of the sum of `n^(-s_(0,k))` over admissible `n` of length `l`; Proposition 7.3: `mu_(m,k) = (mu_(0,k)/mu_(0,0)) mu_(m,0)`, so for `1 < N < b` a pole at `s_(m,k)` exists iff both `s_(m,0)` and `s_(0,k)` are poles; the introduction states the off-real residues are given as a limit and not studied further, and no numerical residue is printed | https://arxiv.org/abs/2602.19727 |
| Burnol 2026 oscillations | The asymptotic oscillations of moments related to Dirichlet series with missing digits, arXiv:2604.24754 - the rescaled moments of the discrete measures used to evaluate missing-digit zeta series numerically are asymptotically 1-periodic in the base logarithm of the index, that is asymptotically invariant under multiplying the index by the base | https://arxiv.org/abs/2604.24754 |
| Allouche, Hu and Morin 2024 | Ellipsephic harmonic series revisited, arXiv:2403.05678 - the `s = 1` endpoint of the same object for a missing digit or a missing block in any base, with the limit `B^(length of w) log B` for the sum of reciprocals of the integers containing exactly `k` occurrences of the block `w`, as `k` goes to infinity. It carries Kempner 1914's statement and bibliographic record verbatim and treats no other value of `s` | https://arxiv.org/abs/2403.05678 |
| Flajolet, Grabner, Kirschenhofer, Prodinger and Tichy 1994 | Mellin transforms and asymptotics: digital sums, Theoret. Comput. Sci. 123(2), 291-314 - the Mellin-Perron treatment of digit sums. For `Phi(n) = sum_(k < n) 2^(v(k))`, the number of odd binomial coefficients in the first `n` rows of Pascal's triangle, `Phi(N)/N^rho` is a periodic function of `log_2 N` with `rho = log_2 3`, and its Fourier coefficients are computed | https://doi.org/10.1016/0304-3975(92)00065-Y |
| Flajolet, Gourdon and Dumas 1994 | Mellin Transforms and Asymptotics: Harmonic Sums, INRIA Research Report RR-2369, read at source in the HAL copy - the dictionary this tree reads pole lattices with: a pole of the Mellin transform at `sigma + i t` with `t` non-zero contributes a term oscillating in `log x` with period `2 pi / t`, and simple poles regularly spaced at `sigma + 2 i k pi / log B`, which arise from a factor `(1 - B^(-s))^(-1)`, contribute `x^(-sigma)` times a Fourier series in `log_B x` | https://inria.hal.science/inria-00074307 |
| Dartyge and Mauduit 2000 | Nombres presque premiers dont l'ecriture en base r ne comporte pas certains chiffres, J. Number Theory 81(2), 270-291, doi:10.1006/jnth.1999.2458 - the ellipsephic almost-prime theorem. ScienceDirect refuses fetchers, so the result is carried from Maynard 2019's restatement read at source: the work of Dartyge and Mauduit shows infinitely many integers of the missing-digit set have at most 2 prime factors, resting on that set being well distributed in arithmetic progressions | https://doi.org/10.1006/jnth.1999.2458 |
| Kim 2024 | The divisor function over integers with a missing digit, arXiv:2411.09076v2 - divisor sums of `d_2` over short intervals of the base-`g` one-missing-digit set, run through Maynard's Fourier bounds; its own framing is that the LACK of multiplicative structure in the set blocks the standard approaches. The nearest published multiplicative function over a missing-digit set, and no Mobius, Liouville or Mertens sum appears in it | https://arxiv.org/abs/2411.09076 |
| Nath 2024 | Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory, J. London Math. Soc. 109(1), e12837, doi:10.1112/jlms.12837, read at source in arXiv:2108.09212v2 - Bombieri-Vinogradov type theorems for `Lambda(n) 1_A(n)` over the base-`b` set `A` missing one digit, `b` large; each saves an arbitrary power of `log X` against the trivial size `X^zeta` with `zeta = log(b-1)/log b`. THE THREE LEVELS ARE NOT INTERCHANGEABLE. Theorem 1 sums `max_((c,d)=1) abs(E(X; d, c))` over `d <= D` and reaches only `D = X^(1/3 - delta)`; Theorem 2 splits the modulus as `d_1 d_2` with `D_1 = X^(1/3 - delta)` and `D_2 = X^(1/9)`, so `X^(4/9 - delta)` in the product, at fixed `c` with no max; Theorem 3 is the only one reaching `D = X^(1/2 - delta)` and it is WEIGHTED, summing `xi(d) E(X; d, c)` against a well-factorable `xi` of level `D`, with no absolute value inside and no max over `c`. The near-`1/2` level is the well-factorable one and is never an unweighted Bombieri-Vinogradov level. The method is the circle method on the Fourier structure of the missing-digit set together with exponential sums over primes in progressions; with the semi-linear sieve it gives upper and lower bounds of the right order for primes `p = 1 + m^2 + n^2` missing a digit in a large odd base. Read at source: Theorems 1-3 (pp.3-4) with the definition of `E(X; d, c; b, r)` (the last digit `r` fixed, `(r,b) = 1`; `b` of order `10^632` at `delta = 1/100` by the paper's own remark); the set-up (pp.7-9), Theorem 7 (pp.40-41) and the proofs of Theorems 1-3 (pp.41-43); Lemmas 9.1-9.4 (pp.45-46), the `l^1`, large-sieve, hybrid and `l^infinity` bounds of the digit transform carried from Maynard's Primes and polynomials with restricted digits. The set enters the proof only through those four norms; every modulus is carried by `Lambda` (Bombieri-Vinogradov in condition (b), exponential sums over primes in progressions in condition (c)); no distribution of `1_A` in progressions is used or proved, the only set-level fact being the last-digit identity `sum_(n < X, n = r mod b) 1_A(n) = X^zeta/(b-1)` for `r` not congruent to `a_0`. A whole-text search of the 54-page source finds `Mobius` and `Liouville` nowhere and `Mertens` twice, both times Mertens' theorem on a product over primes cited to Koukoulopoulos, never the Mertens function | https://arxiv.org/abs/2108.09212 |
| Leng and Sawhney 2025 | Vinogradov's theorem for primes with restricted digits, arXiv:2409.06894v3, read at source - Theorem 1.1: for `g` sufficiently large and any `b` in `{0, ..., g-1}`, every sufficiently large odd `N` is `p_1 + p_2 + p_3` with each `p_i` prime and in `S_b`, the base-`g` integers with no digit equal to `b`. The ternary additive problem is settled on a missing-digit set; the binary one is not. The controlled norm is the `l^1` of the digit transform: the circle integral of `abs(sum_(n < g^k) 1_(n in S_b)(n) e(n theta))` is `<<_eps g^(eps k)` with `eps -> 0` as `g -> infinity`, far past the `g^(k/2)` a square-root heuristic gives, and Lemma 3.2 bounds the same integral for a product measure over digit blocks by `(C log g)^k`. The prime side is a grand zero-density input rather than a level of distribution: the main term swings by a power of `N` across dyadic ranges, so no Fourier transference absorbs it. No Mobius or Mertens sum appears in it | https://arxiv.org/abs/2409.06894 |
| Green 2012 | On (not) computing the Mobius function using bounded depth circuits, Combin. Probab. Comput.; the arXiv page records the acceptance without volume or pages. Any `F` from `{0, ..., N-1}` to `{-1, 1}` computable from the binary digits of `x` by a bounded depth circuit satisfies `E_(0 <= x < N) mu(x) F(x) = o(1)`; the indicator of a set defined by a condition on each binary digit is computable in bounded depth | https://arxiv.org/abs/1103.4991 |
| Baker and Harman 1991 | Exponential Sums Formed with the Mobius Function, J. London Math. Soc. (2) 43(2), 193-198, doi:10.1112/jlms/s2-43.2.193 - the conditional uniform bound on `S(x, theta) = sum_(n <= x) mu(n) e(n theta)`, the statement quoted from the original, pp. 193-194: if `L(s, chi)` has no zeros in the half plane `sigma > a` for every Dirichlet character `chi`, then `max_theta abs(S(x, theta)) << x^(b + eps)` with `b = a + 1/4` on `1/2 <= a < 11/20`, `b = 4/5` on `11/20 <= a < 3/5`, `b = (a + 1)/2` on `3/5 <= a < 1`, the implied constants depending at most on `eps`; in particular under the generalized Riemann hypothesis the exponent falls from `5/6 + eps` to `3/4 + eps`, and a Liouville analogue is stated. The same pages carry the sharper single-arc Proposition `S(x, theta) << x^(a + eps) q^(1/2) (1 + x abs(theta - r/q))^(1/2)`. Wiley answers automated fetches with a 403, so the bibliographic record is resolved on the Crossref entry for the DOI, and the `3/4 + eps` statement is restated independently in the Zhang 2024 and Porritt 2018 rows | https://doi.org/10.1112/jlms/s2-43.2.193 |
| Zhang 2024 | On an exponential sum related to the Mobius function, Proc. Amer. Math. Soc. 152(4), 1373-1376, doi:10.1090/proc/16270, read at source in arXiv:2204.04613v2 - the sharpening of Baker and Harman 1991 on its first range. Its Theorem 1.1: under the same hypothesis, that `L(s, chi)` has no zeros in `sigma > a` for every Dirichlet character `chi`, `max_alpha abs(S(x, alpha)) << x^(b + eps)` with `b = (8a - 7a^2)/(4 - 2a)` for `a` in `[1/2, 4/7]`. The paper prints `1/4 + a >= (8a - 7a^2)/(4 - 2a)` on that range with equality only at `a = 1/2`, so the new exponent is never worse and is better inside. At `a = 1/2` it is exactly `3/4`, so the GRH endpoint is not moved. The AMS text is paywalled; the volume, issue and pagination are resolved on the Crossref entry for the DOI | https://arxiv.org/abs/2204.04613 |
| Porritt 2018 | A note on exponential-Mobius sums over `F_q[t]`, Finite Fields Appl. 51, 298-305, doi:10.1016/j.ffa.2018.02.005 - the function field companion, the statement quoted from the original, pp. 298-299: it restates the Baker and Harman theorem, records that the conjectured exponent for `max_theta abs(sum_(n <= x) mu(n) e(n theta))` is `1/2` in place of `3/4`, and carries Davenport's unconditional `max_theta abs(sum_(n <= x) mu(n) e(n theta)) <<_A x (log x)^(-A)`. The polynomial analogue of the `3/4 + eps` bound is deduced from Weil's Riemann hypothesis for curves over a finite field, which is a theorem there and not a hypothesis, on the approach of Hayes to exponential sums over irreducible polynomials. ScienceDirect refuses fetchers; the open copy is arXiv:1711.08729v2, whose abstract states the Baker and Harman bound and that deduction | https://doi.org/10.1016/j.ffa.2018.02.005 |
| Titchmarsh 1986 | The Theory of the Riemann Zeta-Function, second edition revised by D. R. Heath-Brown, Clarendon Press, ISBN 0-19-853369-1 - Theorem 14.25 (C) at pages 369-370: a necessary and sufficient condition for the Riemann hypothesis is `M(x) = O(x^(1/2 + eps))`, with Theorem 14.25 (B) the companion equivalence that `sum mu(n) n^(-s)` converging on `Re s > 1/2` is necessary and sufficient too. The canonical citation for the Mertens-RH equivalence. The book has no DOI; the url is the scan the theorem is read in | https://sites.math.rutgers.edu/~zeilberg/EM18/TitchmarshZeta.pdf |
| Wolfram 1983 | Statistical mechanics of cellular automata, Rev. Mod. Phys. 55(3), 601-644; the table partitioning the 256 elementary rules into 88 classes under reflection and state complementation | https://doi.org/10.1103/RevModPhys.55.601 |
| Li and Packard 1990 | The structure of the elementary cellular automata rule space, Complex Systems 4(3), 281-297; the derivation of the 88 count. The only full text reached is a scan, so its statements here are carried from Schaller and Svozil's restatement | https://www.complex-systems.com/abstracts/v04_i03_a03/ |
| Martinez 2013 | A note on elementary cellular automata classification, arXiv:1306.5577; prints the full 88-row cluster table and names the transformations reflection, negation and complementation | https://arxiv.org/abs/1306.5577 |
| Schaller and Svozil 2025 | Irreducible rules and equivalence classes of one-dimensional cellular automata, arXiv:2512.08117; state permutations and lattice isometries generate the symmetry group, extended by neighbourhood scaling, and free permutation of the three neighbourhood coordinates is not among them | https://arxiv.org/abs/2512.08117 |
| Harrison 1963 | The number of transitivity sets of Boolean functions, J. Soc. Indust. Appl. Math. 11(3), 806-828, doi:10.1137/0111059, the JSTOR page beside answering automated fetches with a bot wall; the cycle-index formula OEIS credits for A000616, the 22 classes of three-variable Boolean functions under the order-48 hyperoctahedral group | https://www.jstor.org/stable/2946322 |
| Harrison 1963 IEEE | The number of equivalence classes of Boolean functions under groups containing negation, IEEE Trans. Electron. Comput. 12, 559-561; the source OEIS credits for A000370, the 14 NPN classes | https://doi.org/10.1109/PGEC.1963.263656 |
| Langton 1990 | Computation at the edge of chaos: phase transitions and emergent computation, Physica D 42(1-3), 12-37; lambda is the fraction of rule-table entries mapping to a non-quiescent state. Every copy reached is a scan, so the definition is carried from restatements | https://doi.org/10.1016/0167-2789(90)90064-V |
| Mitchell, Hraber and Crutchfield 1993 | Revisiting the edge of chaos: evolving cellular automata to perform computations, Complex Systems 7(2), 89-130; the abstract states the earlier interpretation of lambda's role is not correct | https://www.complex-systems.com/abstracts/v07_i02_a01/ |
| Wuensche 1999 | Classifying cellular automata automatically, Complexity 4(3), 47-66; Z is the probability that the next unknown cell in a partial pre-image is uniquely determined | https://doi.org/10.1002/(SICI)1099-0526(199901/02)4:3<47::AID-CPLX9>3.0.CO;2-V |
| Martin, Odlyzko and Wolfram 1984 | Algebraic properties of cellular automata, Comm. Math. Phys. 93(2), 219-258; rule 90 is the Laurent polynomial x + x^-1 over GF(2) and rule 150 is x + 1 + x^-1. It observes rule 90's single-cell pattern has fractal dimension log2 3; it does not prove it and never says Pascal | https://doi.org/10.1007/BF01223745 |
| Willson 1984 | Cellular automata can generate fractals, Discrete Applied Mathematics 8(1), 91-99. Paywalled; only the bibliographic record was read | https://doi.org/10.1016/0166-218X(84)90082-9 |
| Willson 1987 | Computing fractal dimensions for additive cellular automata, Physica D 24(1-3), 190-206; the abstract states the largest eigenvalue of an integer matrix built from the automaton gives the fractal dimension. Paywalled; the body was not read, so no rule-150 number is drawn from it | https://doi.org/10.1016/0167-2789(87)90074-1 |
| Cook 2004 | Universality in elementary cellular automata, Complex Systems 15(1), 1-40; rule 110 is Turing universal | https://www.complex-systems.com/abstracts/v15_i01_a01/ |
| Culik and Yu 1988 | Undecidability of CA classification schemes, Complex Systems 2(2), 177-190; the Wolfram classes are made formal and membership is undecidable | https://www.complex-systems.com/abstracts/v02_i02_a02/ |
| Ollinger 2009 | Intrinsically universal cellular automata, EPTCS 1, 199-204; intrinsic universality is undecidable though recursively enumerable, rule 110's Turing universality does not give intrinsic universality, and whether rule 110 is intrinsically universal is open | https://doi.org/10.4204/EPTCS.1.19 |
| Delorme, Mazoyer, Ollinger and Theyssier 2011a | Bulking I: an abstract theory of bulking, Theoret. Comput. Sci. 412(30), 3866-3880 | https://doi.org/10.1016/j.tcs.2011.02.023 |
| Delorme, Mazoyer, Ollinger and Theyssier 2011b | Bulking II: classifications of cellular automata, Theoret. Comput. Sci. 412(30), 3881-3905; bulking is a quasi-order comparing space-time diagrams up to rescaling, not a tensor operation on neighbourhood masks | https://doi.org/10.1016/j.tcs.2011.02.024 |
| Amoroso and Patt 1972 | Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, J. Comput. System Sci. 6(5), 448-464; both properties are decidable in one dimension | https://doi.org/10.1016/S0022-0000(72)80013-8 |
| Sutner 1991 | De Bruijn graphs and linear cellular automata, Complex Systems 5(1), 19-30; quadratic-time tests for reversibility and surjectivity | https://www.complex-systems.com/abstracts/v05_i01_a03/ |
| Sloane 2015 | On the number of ON cells in cellular automata, arXiv:1503.01168; for an odd-rule automaton with neighbourhood F the state at generation n is F^n and the ON count is the number of its nonzero terms. Fredkin's Replicator is the eight-cell Moore neighbourhood, (1/x+1+x)(1/y+1+y) - 1, which is not separable | https://arxiv.org/abs/1503.01168 |
| Ekhad, Sloane and Zeilberger 2015 | Odd-rule cellular automata on the square grid, arXiv:1503.04249; the exhaustive sweep of planar odd-rule neighbourhoods that A246035 cites as OddRule 777 | https://arxiv.org/abs/1503.04249 |
| Eppstein 2010 | Growth and decay in Life-like cellular automata, arXiv:0911.2890; a four-way classification of semi-totalistic planar rules by escape and extinction. It proves no rule universal | https://arxiv.org/abs/0911.2890 |
| Rendell 2016 | Turing Machine Universality of the Game of Life, Springer; a universality proof by direct Turing-machine construction rather than the counter machine of the classical argument | https://doi.org/10.1007/978-3-319-19842-2 |
| Cox, Sederberg and Chen 1998 | The moving line ideal basis of planar rational curves, Computer Aided Geometric Design 15(8), 803-827, doi:10.1016/S0167-8396(98)00014-4 - the mu-basis degree identity `mu_1 + mu_2 = n - deg(gcd)` that [DISCOVERIES](/research/discoveries/) records as the prior art behind `delta_1 + delta_2 = 12R + 5`, so neither that identity nor the rank-2 freeness under it is claimable. Title, journal, volume, issue, pages and year are resolved at the Crossref record; ScienceDirect refuses fetchers, so the text is not read here and the Index Sum Theorem naming on that ledger line is carried unchecked | https://doi.org/10.1016/S0167-8396(98)00014-4 |
| Burch 1968 | On ideals of finite homological dimension in local rings, Math. Proc. Cambridge Philos. Soc. 64(4), 941-948, doi:10.1017/S0305004100043620 - the Hilbert-Burch theorem, which [DISCOVERIES](/research/discoveries/) leans on for rank-2 freeness of the truncated first syzygy module of the carry core. Resolved at the Crossref record, whose title carries a stray accent in `homological`. The textbook statement is Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, GTM 150, doi:10.1007/978-1-4612-5350-1, whose Crossref record gives the short title only and whose Springer page sits behind an automated challenge, so its theorem number is not checked here | https://doi.org/10.1017/S0305004100043620 |
| Kuipers and Niederreiter 1974 | Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974, reprinted Dover 2006, ISBN 0-471-51045-9 - the source of the two discrepancy inequalities an orbit argument runs on. The scan is lend-only and its full-text search answers automated fetches with a 403, so nothing is read at source. Chapter 2 Theorem 2.5, the Erdos-Turan inequality with explicit constants, is quoted verbatim in Grozdanov and Stoilova, The Inequality of Erdos-Turan-Koksma: Walsh and Haar Functions Over Finite Groups, Math. Balkanica (N.S.) 19, Fasc. 3-4, 349-366, read at source at p.350: `In Kuipers and Niederreiter [13, theorem 2.5] it is shown that for an arbitrary net xi_N = {x_0, ..., x_(N-1)} of N >= 1 points in [0,1) the inequality D(xi_N) <= 6/(m+1) + (4/pi) sum_(k=1)^(m-1) (1/k - 1/m) abs((1/N) sum_(j=0)^(N-1) exp(2 pi i k x_j)) holds for each integer m >= 1`. THE PRINTED WEIGHT IS `1/k - 1/m` OVER `k <= m - 1`, NOT `1/k` OVER `k <= m`; since `1/k - 1/m <= 1/k` the looser form is implied and not equal. The constants `6` and `4/pi` are confirmed, and the quantity bounded is the extreme discrepancy `D_N`, not the star discrepancy. Koksma's inequality, `abs((1/N) sum_(n <= N) f(x_n) - int_0^1 f) <= V(f) D*_N` for `f` of bounded variation on `[0,1]`, sits in the same Chapter 2 Section 5 on numerical integration; the same section's Theorem 5.5, the Hardy-Krause Koksma-Hlawka inequality, is cited at source in arXiv:2207.11840 as `[32, Theorem 5.5]`, which places Koksma's inequality at Theorem 5.1 of that section without verifying the number or the star | https://archive.org/details/uniformdistribut0000kuip |
| Bourgain 2005 | Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal. 15, 1-34, doi:10.1007/s00039-005-0500-4. Springer answers automated fetches with a login redirect and withholds even the abstract, so the statement is carried from the author's own announcement note, New bounds on exponential sums related to the Diffie-Hellman distributions, C. R. Math. Acad. Sci. Paris 338(11), 825-830, doi:10.1016/j.crma.2004.03.027, read at source. Theorem 2.1, p.828, verbatim: `Given delta > 0, there is delta' > 0 such that if theta in F_p^* is of multiplicative order t and t >= t_1 > p^delta, then max_(a in F_p^*) abs(sum_(s=1)^(t_1) e_p(a theta^s)) < t_1 p^(-delta')`. So `p` IS PRIME, the saving is `N p^(-delta)` and NOT `N^(1-delta)`, the summation length runs `p^delta < t_1 <= t = ord_p(theta)` with the lower bound strict, the coefficient condition is `a` invertible mod `p`, and `delta'` depends on `delta` alone. Theorem 3.1, p.829, is the same saving against the von Mangoldt weight: `max_(a in F_p^*) abs(sum_(n=1)^N Lambda(n) e_p(a theta^n)) < N p^(-delta')` for `t > p^delta` and `N > t^(2+delta)`. Theorems 2.2-2.4 carry the double and incomplete double sums `sum_(s') abs(sum_s e_p(a theta^s + c theta^(s s')))` with the same shape of saving | https://doi.org/10.1007/s00039-005-0500-4 |
| Barban, Linnik and Chudakov 1964 | On prime numbers in an arithmetic progression with a prime-power difference, Acta Arith. 9, 375-390, doi:10.4064/aa-9-4-375-390. The free scan answers automated fetches with a 403, so nothing is read at source; the result is carried from the restatement in arXiv:2107.04348, read at source: `For any fixed odd prime number p, Barban, Linnik and Chudakov proved that an asymptotic of the form pi(x, p^N, a) = (1/phi(p^N)) int_2^x dt/log t (1 + O(1/L^A)) holds for single residue classes modulo a prime power p^N in the wider range p^N <= x^(3/8-eps)`. So the modulus range is `p^N <= x^(3/8-eps)` with `p` a FIXED odd prime, and the saving is an arbitrary power of `log x`. The original theorem numbering is not read | https://doi.org/10.4064/aa-9-4-375-390 |
| Gallagher 1972 | Primes in progressions to prime-power modulus, Invent. Math. 16, 191-201, doi:10.1007/BF01425492. Springer answers automated fetches with a login redirect and the archive scan carries no text layer, so nothing is read at source. The zero-free region is quoted, jointly with Iwaniec 1974, in arXiv:1105.3895, read at source: `the restriction (0.1) follows from results of Gallagher and Iwaniec ([G], [I]) that provide the zero-free region 1 - sigma < c_1 (log qT . log log qT)^(-3/4), abs(gamma) < T where rho = sigma + i gamma, for special moduli q that are powers of a fixed integer (here q = 2^j)`, and that paper adds `this region is larger than what's available in the general case`. The modulus range of the prime-counting consequence is quoted in arXiv:2107.04348, read at source: `the exponent 3/8 was improved to 2/5 by Gallagher`, that is `p^N <= x^(2/5-eps)`. The original theorem numbering is not read | https://doi.org/10.1007/BF01425492 |
| Iwaniec 1974 | On zeros of Dirichlet's L series, Invent. Math. 23, 97-104, doi:10.1007/BF01405163. Springer answers automated fetches with a login redirect, so nothing is read at source. Its Theorem 2 is restated in Banks and Shparlinski, Bounds on short character sums and L-functions for characters with a smooth modulus, arXiv:1605.07553, read at source: `Iwaniec [7, Theorem 2] yields a similar bound with (log q(abs(t)+3))^(3/4) (log log q(abs(t)+3))^(3/4) in the denominator`, that is a zero-free region whose width is `c / ((log q(abs(t)+3))^(3/4) (log log q(abs(t)+3))^(3/4))`. THE `2/3` EXPONENT IS NOT IWANIEC'S: it belongs to that same paper's own Theorem 3.2, `there is a constant A > 0 such that if theta = A/((log q)^(2/3) (log log q)^(1/3)), then there exists at most one primitive character chi modulo q such that L(s, chi) has a zero in the region {s in C : sigma > 1 - theta, abs(t) <= q^C}`, which improves Iwaniec. Iwaniec's own exponents are `3/4` on both logarithms, matching the region arXiv:1105.3895 attributes jointly to Gallagher and Iwaniec. In every restatement reached the exceptional real zero is excepted, never excluded | https://doi.org/10.1007/BF01405163 |
| Montgomery and Vaughan 2007 | Multiplicative Number Theory I: Classical Theory, Cambridge Studies in Advanced Mathematics 97, ISBN 978-0-521-84903-6, read at source - the numbering the classical Mobius inputs actually carry. Throughout, `tau = abs(t) + 4`. Theorem 6.6, p.172: `There is an absolute constant c > 0 such that zeta(s) != 0 for sigma >= 1 - c/log tau`, the de la Vallee Poussin region. Theorem 6.7 is NOT that region but the `zeta'/zeta << log tau` bound inside it, and Theorem 6.9 is the quantitative prime number theorem `psi(x) = x + O(x exp(-c sqrt(log x)))`. THE TWO MOBIUS BOUNDS ARE NOT A NUMBERED THEOREM: they are equations (6.17) and (6.18), p.182, printed as prose after the proof of Theorem 6.9 under the words `it may be shown that` - `M(x) = sum_(n <= x) mu(n) << x exp(-c sqrt(log x))` for `x >= 2`, and `sum_(n <= x) mu(n)/n << exp(-c sqrt(log x))`, `since 1/(s zeta(s+1)) is analytic at s = 0`, whence `sum_(n=1)^infinity mu(n)/n = 0`. Exercise 17 of Section 6.2 carries the coprime version UNIFORMLY IN THE MODULUS: `if q <= x then (a) sum_(n <= x, (n,q) = 1) mu(n)/n << exp(-c sqrt(log x))` and `(b) sum_(n <= x, (n,q) = 1) mu(n) log(n)/n = -q/phi(q) + O(exp(-c sqrt(log x)))`, both by elementary reasoning from (6.18). Theorem 11.3, p.360: `There is an absolute constant c > 0 such that if chi is a Dirichlet character modulo q, then the region R_q = {s : sigma > 1 - c/log q tau} contains no zero of L(s, chi) unless chi is a quadratic character, in which case L(s, chi) has at most one, necessarily real, zero beta < 1 in R_q`; such a zero is called exceptional. Corollary 11.8, p.368, is Landau's: `prod_chi L(s, chi) has at most one zero in the region sigma > 1 - c/log q tau`, the product over all characters mod q, the zero necessarily real and its character quadratic. Corollary 11.10 is Page's, the same conclusion for `prod_(q <= Q) prod*_chi L(s, chi)` in `sigma >= 1 - c/log Q tau` with an ABSOLUTE constant, which is stronger than a `c(Q)`. Corollary 11.19 is the Siegel-Walfisz theorem, for `psi(x; q, a)` at `q <= (log x)^A`. THE MOBIUS FORM IS AN EXERCISE, NOT A COROLLARY: Exercise 8 of Section 11.3, p.384, `Let c_1 be the constant in Theorem 11.16, and suppose that A is given, A > 0. Show that if q <= (log x)^A and chi is a character modulo q, then ... M(x, chi) <<_A x exp(-c_1 sqrt(log x))`, with `M(x, chi) = sum_(n <= x) chi(n) mu(n)` defined at (11.39), p.383, alongside the Liouville companion `Lambda(x, chi) = sum_(n <= x) chi(n) lambda(n)`. THE SAVING IS EXPONENTIAL, `exp(-c_1 sqrt(log x))`, not the `(log y)^(-A)` shape usually quoted. Nothing in the book states the fixed-modulus fact, that a finite family of real characters of conductor dividing a fixed `Q` has a genuinely zero-free `sigma > 1 - c(Q)/log tau`, as a numbered result; Corollary 11.10 is the printed statement nearest to it | https://www.cambridge.org/core/books/multiplicative-number-theory-i/4E45519B26115AEEA4839C6C38206ACD |
| Maynard 2022 | Primes and polynomials with restricted digits, Int. Math. Res. Not. IMRN 2022(14), 10626-10648, doi:10.1093/imrn/rnab002, read at source in the published text; the arXiv copy is the 2015 first version, arXiv:1510.07711. No Type I or Type II estimate is in it: the strings `Type I` and `Type II` name no result, `Vaughan` and `Vinogradov` name no lemma, and the introduction says plainly `Somewhat surprisingly, the Fourier structure is sufficient to deduce the existence of primes in A using only existing exponential sum estimates for the primes, and without having to investigate further bilinear sums`. Type I-II sums appear twice, as prospects only, `it appears that the method of bilinear sums, Harman's sieve and zero density estimates all have the potential to show the existence of primes missing digits when the base is noticeably smaller` and `one would hope that utilizing Type I-II sums and Harman's sieve would extend this to sets of smaller density`. Theorem 1.1, p.2: `Let q > 2000000, a_0 in {0, ..., q-1} and A = {sum_(i >= 0) n_i q^i : n_i in {0, ..., q-1} minus {a_0}}`, then for any `A > 0`, `sum_(n < q^k) Lambda(n) 1_A(n) = kappa_q(a_0) (q-1)^k + O_A((q-1)^k (log q^k)^(-A))` with `kappa_q(a_0) = q/(q-1)` if `(a_0, q) != 1` and `q(phi(q)-1)/((q-1)phi(q))` if `(a_0, q) = 1`; the remark adds `a more involved calculation shows that q > 2500 is sufficient by the same method` and `one might conjecture that the result would remain true for all q > 2`. Theorem 1.2 needs `q > exp(exp(2r))` for a degree-`r` polynomial. Theorem 1.3 takes `0 < s < q^(1/5-eps)` excluded digits, or `q - s >= q^(4/5+eps)` when they are consecutive, and the `4/5` is the paper's own limit: `the exponent 4/5 is ultimately related to the 4/5 exponent of Lemma 4.2 for an exponential sum over primes and represents a limit of our basic method`. Lemma 4.2, p.7, is that whole prime input: for `alpha = a/d + beta` with `(a,d) = 1` and `abs(beta) < 1/d^2`, `sum_(n < x) Lambda(n) e(n alpha) << (x^(4/5) + x^(1/2) abs(d beta)^(-1/2) + x abs(d beta)^(1/2)) (log x)^4`. The digit set enters only through Section 5's four Fourier norms, Lemma 5.1 the `l^1` bound `sup_theta sum_(0 <= a < q^k) abs(F_(q^k)(theta + a/q^k)) << (C_q q log q)^k` with `C_q in [1/log q, 1 + 3/log q]`, Lemma 5.2 the large sieve, Lemma 5.3 the hybrid bound, complete at source: `Let B, D >> 1. Then sum_(d ~ D) sum_((l,d)=1) sum_(abs(eta) < B, q^k l/d + eta in Z) abs(hat F_(q^k)(l/d + eta/q^k)) << (q-1)^k (D^2 B)^(alpha_q) + D^2 B (C_q log q)^k`, `alpha_q = log(C_q (q/(q-1)) log q)/log q`, WITH NO COPRIMALITY CONDITION `(d,q) = 1` AND NO RANGE ON `D` BEYOND `D >> 1`; and Lemma 5.4 the `l^infinity` bound `(q-1)^k exp(-c_q k/log d)` at `d < q^(k/3)`, `d = d_1 d_2`, `(d_1,q) = 1`, `d_1 != 1`, `abs(eps) < 1/(2 q^(2k/3))`, `(l,d) = 1`, for `some constant c_q > 0 depending only on q` - NO SIZE FOR `c_q` IS GIVEN, and the proof runs on `abs(e(n theta) + e((n+1) theta))^2 = 2 + 2 cos(2 pi theta) < 4 exp(-2 norm(theta)^2)`, so it consumes two consecutive allowed digits. Section 9, p.17, is the multi-digit extension and is a sketch by its own words, `leaving the precise details to the interested reader`: `C_(q,s) = 1 + (2+s)/log q` in general and `2 + 2/log q` when the excluded digits are consecutive, `alpha_(q,s) = log(C_(q,s) (q/(q-s)) log q)/log q`, `Lemma 5.2 remains unchanged whilst in Lemma 5.3 all occurrances of q - 1 should be replaced by q - s`, and then `Lemmas 5.4, 6.1, 6.2, 7.1, 7.2, 7.3 all go through as before`, with `alpha_(q,s) <= log s/log q + eps` at `s < q/2` and `alpha_(q,s) <= log(q/(q-s))/log q + eps` when consecutive. The paper also records that `since these estimates are only used when the modulus is highly composite, in fact Siegel zeros do not play a role, and so the error terms could be replaced by effective ones of size O((q-1)^k exp(-c k^(1/2)))`. No Mobius or Mertens sum appears in it | https://doi.org/10.1093/imrn/rnab002 |
| Heath-Brown 1982 | Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34(6), 1365-1377, doi:10.4153/CJM-1982-095-9. The publisher's scan carries no text layer, so nothing is read at source; the identity is carried from three later papers, each read at source and each citing this one, and ALL THREE STATE IT FOR `Lambda` ONLY. arXiv:1112.0201 cites it as `[2, Lemma 1]`: `if n <= X and J is a positive integer, then Lambda(n) = sum_(j=1)^J binom(J,j) (-1)^j sum_(n = n_1 ... n_(2j), n_1, ..., n_j <= X^(1/J)) mu(n_1) ... mu(n_j) log(n_(2j))`. arXiv:math/0412227 prints the `J = 10` case and arXiv:1402.0811 the convolution form `Lambda = sum_(j=1)^K (-1)^(j-1) binom(K,j) mu_(<=)^(*j) * 1^(*(j-1)) * L` with `mu_(<=)(n) = mu(n) 1_(n <= (2x)^(1/K))`. The validity range is `n <= z^K` in every restatement. THE MOBIUS FORM IS NOT IN THIS PAPER; its citable home is Iwaniec and Kowalski equation (13.38). The lemma number `1` rests on the citation convention of arXiv:1112.0201, not on the printed page | https://doi.org/10.4153/CJM-1982-095-9 |
| Iwaniec and Kowalski 2004 | Analytic Number Theory, Amer. Math. Soc. Colloquium Publications 53, doi:10.1090/coll/053 - the home of the MOBIUS form of the Heath-Brown identity, Chapter 13 equation (13.38): `mu(n) = - sum_(1 <= k <= K) (-1)^k binom(K,k) sum_(m_1 ... m_k n_1 ... n_(k-1) = n, m_1, ..., m_k <= u) mu(m_1) ... mu(m_k)` for `K >= 1`, `n >= 1`, `u >= n^(1/K)`; folding the sign gives `mu = sum_(k=1)^K (-1)^(k-1) binom(K,k) mu_(<= u)^(*k) * 1^(*(k-1))`. The `Lambda` form is a separate numbered statement, Proposition 13.3. Not read at source; the equation number, the statement and the range are carried from arXiv:2101.08773, read at source, which cites `[IK04], eq. (13.38)` and records a typo in the printed summation condition | https://doi.org/10.1090/coll/053 |
| Lalley 1989 | Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits, Acta Math. 163, 1-55. For `N(a,x) = sum_n sum_(sigma^n y = x) g(y) 1{S_n f(y) <= a}`: "Say that f is a lattice function if f is cohomologous to a function taking values in a discrete subgroup of R; otherwise, say that f is a nonlattice function." Proposition 2.1 gives the unique `delta > 0` with `lambda_(-delta f) = 1`; Theorem 1: `f` nonlattice gives `N(a,x) ~ C(x) e^(a delta)`; Theorem 2: `f` integer-valued gives `N(a,x) ~ C(x) e^([a] delta)`, the log-periodic case. The dichotomy is a property of `f` alone, whichever variable is fed in | https://doi.org/10.1007/BF02392732 |
| Daubechies 1988 | Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41(7), 909-996. Abstract: "We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width." Equation (2.15) is the two-scale relation `phi(x) = sum_(n=0)^N c_n phi(2x - n)` with `supp phi` in `[0, N]`; the remark after (4.29) states that a `C^k` solution with that support forces `k <= N - 2`, so regularity is bought only by widening the mask past one residue box | https://doi.org/10.1002/cpa.3160410705 |
| Daubechies and Lagarias 1992 | Two-scale difference equations II: local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23(4), 1031-1079. Page 1036 rewrites `phi(x) = sum_(n=0)^N c_n phi(2x - n)` as `Phi(x) = T_0 Phi(2x)` on `[0, 1/2]` and `T_1 Phi(2x - 1)` on `[1/2, 1]`, `Phi = (phi(x), ..., phi(x + N - 1))`, `T_0 = (c_(2i-j-1))`, `T_1 = (c_(2i-j))`, `1 <= i, j <= N`, under `sum c_(2n) = sum c_(2n+1) = 1`; the Holder exponent is `-log_2` of the joint spectral radius on the common invariant subspace. Quoted at source in Dumas, arXiv:0807.1523, Section 2.4 | https://doi.org/10.1137/0523084 |
| Rota and Strang 1960 | A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22, 379-381. For a finite family `F` of matrices and any matrix norm, `jsr(F) = lim_k sup_(P in P_k(F)) norm(P)^(1/k)`, independent of the norm, and the joint spectral radius equals the infimum over all matrix norms of the largest norm of a member of `F`; bibliographic data and that statement verbatim in Blondel, The birth of the joint spectral radius, Linear Algebra Appl. 428 (2008) 2261-2264 | https://doi.org/10.1016/S1385-7258(60)50046-1 |
| Blondel and Nesterov 2005 | Computationally efficient approximations of the joint spectral radius, SIAM J. Matrix Anal. Appl. 27, 256-272. Equation (1.3) brackets the joint spectral radius by `max rho(A_sigma)^(1/k)` below and `max norm(A_sigma)^(1/k)` above. Theorem 1: for a family leaving a proper cone invariant, `(1/m) rho(sum A_i) <= rho(A_1..A_m) <= rho(sum A_i)`. Theorem 2: `rho(A_1^(x)l, ..., A_m^(x)l) = rho^l(A_1..A_m)`. Theorem 3: for such a family, `(1/m^(1/k)) rho^(1/k)(A_1^(x)k + ... + A_m^(x)k) <= rho(A_1..A_m) <= rho^(1/k)(A_1^(x)k + ... + A_m^(x)k)`, and the limit in `k` is the joint spectral radius | https://arxiv.org/abs/math/0407485 |
| Guglielmi and Protasov 2013 | Exact computation of joint spectral characteristics of linear operators, Found. Comput. Math. 13, 37-97. Definition 1: a norm is extremal for `M` if `norm(A_j x) <= rho-hat norm(x)` for all `x`, equivalently `max_j norm(A_j) = rho-hat`. Definition 2: a product `Pi` in `M^n` is a spectrum maximizing product if `rho(Pi)^(1/n) = rho-hat(M)`. The polytope algorithm seeks an extremal polytope norm, characterized by `A_j P subset rho_l P`, under the standing hypothesis stated outright: "in the sequel of this section we assume that M is irreducible", since JSR computation "has to be considered only for irreducible families of matrices" | https://arxiv.org/abs/1106.3755 |
| Bousch and Mairesse 2002 | Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc. 15, 77-111. Defines the greatest Liapunov exponent `lambda^+(A) = sup_(abs(w) >= 1) (1/abs(w)) log rho(A_w)` and states the Lagarias-Wang finiteness conjecture as the assertion that this supremum is always attained. Section 4.1: "We shall prove in the next section that this conjecture is false, by constructing a linear IFS A, consisting of two 2x2 matrices, with nonnegative coefficients, and such that for all w != e, (1/abs(w)) log rho(A_w) < lambda^+(A)" | https://doi.org/10.1090/S0894-0347-01-00380-0 |
| Hare, Morris, Sidorov and Theys 2011 | An explicit counterexample to the Lagarias-Wang finiteness conjecture, Adv. Math. 226, 4667-4701. States the conjecture as "every finite set of real d x d matrices satisfies the finiteness property", the finiteness property being the existence of a periodic product attaining the joint spectral radius; records that Bousch and Mairesse, Blondel, Theys and Vladimirov, and Kozyakin proved counterexamples exist, and supplies the first completely explicit one | https://arxiv.org/abs/1006.2117 |
| Huxley 2003 | Exponential sums and lattice points III, Proc. London Math. Soc. 87(3), 591-609. The lattice-point discrepancy of a planar domain bounded by a piecewise smooth curve is `O(R^K (log R)^Lambda)` with `K = 131/208 = 0.6298076923` in the maximum radius of curvature `R`, improving `46/73` from paper II; the Dirichlet divisor exponent becomes `K/2 = 131/416`. This is the exponent to quote for the Gauss circle problem | https://doi.org/10.1112/S0024611503014485 |
| Bourgain and Watt 2017 | Mean square of zeta function, circle problem and divisor problem revisited, arXiv:1709.04340, the source of the circle exponent `517/824 = 0.6274271845`. WITHDRAWN: the authors' comment records a gap in the proofs of Propositions 2 and 3 and a further problem with Proposition 1', concluding "Theorems 1, 2 and 3 lose their status as theorems"; `517/824` is never quoted as a theorem | https://arxiv.org/abs/1709.04340 |
| Heath-Brown 1999 | Lattice points in the sphere, in Number Theory in Progress (de Gruyter), 883-892. With `S(R) = #{x in Z^3 : abs(x) <= R}`: "Theorem. For any `eps > 0` we have `S(R) = (4/3) pi R^3 + O_eps(R^(21/16+eps))`", sharpening `29/22` of Chamizo and Iwaniec and the `4/3` of Chen and Vinogradov; the error is known to be `Omega(R (log R)^(1/2))` and conjectured `O_eps(R^(1+eps))` | https://ora.ox.ac.uk/objects/uuid:4b17126d-c3a0-4827-8fa2-0ad82872d17e |
| Gauss circle survey 2023 | Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles, arXiv:2305.03549. States the normalisation `N(R) = pi R^2 + Delta(R)` in the radius, Hardy's conjecture `Delta(R) = O(R^(1/2 + o(1)))` and Hardy's 1916 lower bound `Delta(R) != O(R^(1/2) (log R)^(1/4))`; it attributes the best upper bound to Bourgain and Watt, which the withdrawal above supersedes | https://arxiv.org/abs/2305.03549 |
| Van Loan and Pitsianis 1993 | Approximation with Kronecker products, in Linear Algebra for Large Scale and Real-Time Applications, NATO ASI Series 232, Springer. The nearest Kronecker product `min norm(A - B (x) C)_F` is the rank-one SVD of a rearrangement `R(A)` whose rows are the vectorised blocks, `R(B (x) C) = vec(B) vec(C)^T`, so the singular values of `R(A)` are the data's Kronecker spectrum. The chapter is paywalled: the operator and the theorem are read in the verbatim restatement of KoPA: Automated Kronecker Product Approximation, arXiv:1912.02392, which defines `R` and attributes the SVD connection to this chapter | https://doi.org/10.1007/978-94-015-8196-7_17 |
| Cawley and Mauldin 1992 | Multifractal decompositions of Moran fractals, Adv. Math. 92, 196-236. For a self-similar measure under the open set condition, `beta(q)` is defined by `sum_i p_i^q r_i^(beta(q)) = 1` and the multifractal spectrum is its Legendre transform, `f_mu(alpha) = beta*(alpha)` for all `alpha >= 0`, `phi*(x) = inf_y (x y + phi(y))`. Quoted at source in Olsen, Multifractal tubes, arXiv:1307.5223, Section 3, equations (3.9), (3.10) and (2.5); the original is behind a publisher wall and was not opened | https://doi.org/10.1016/0001-8708(92)90064-R |
| Kombrink, Pearse and Winter 2016 | Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable, Math. Z. 283(3), 1049-1070, read at source in arXiv:1501.03764v1: Theorem 1.1(ii) restated as Theorem 3.4 (lattice, OSC, non-integer dimension, a strong feasible open set with the projection condition, pluriphase with respect to Gamma(O)), Definition 2.9 pluriphase, Figure 1's caption naming the carpet tiling with the open square as monophase with bd O in F | https://arxiv.org/abs/1501.03764 |
| Lapidus, Pearse and Winter 2011 | Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. Math. 227, read at source in arXiv:1006.3807: the tube formula and the definitions of generator and pluriphase; Section 6's figure caption calls the Menger sponge generator neither convex nor pluriphase | https://arxiv.org/abs/1006.3807 |
| Lapidus, Pearse and Winter 2013 | Minkowski measurability results for self-similar tilings and fractals with monophase generators, Contemp. Math. 600, 185-204, read at source in arXiv:1104.1641: the monophase case under bd O in F | https://arxiv.org/abs/1104.1641 |
| Karwatowski, base 9 | Digits of primes in base b = 9, 17 pages, preprint on the author's page at the Mathematical Institute of Heinrich Heine University Duesseldorf, undated in the text, read at source: Theorem 1 (p.3) `#{p in A(X)} asymp #A(X)/log X = X^(log 8/log 9)/log X` for base 9 and excluded digit `a_0 in {0, 8}`; the constants (p.5) `lambda(1,4) <= 9^0.3219` replacing `27/77` and `50/77` by `0.3219` and `0.6781`, `lambda(3/2, 4) <= 9^0.14355` replacing `235/154` and `59/433` by `3/2` and `0.14355`, `v = 0.28711` replacing `23/80` (p.7), all by a Mathematica code `handed out on request`; Section 6 (p.16) the criterion for a pair `(b, a_0)`: `g(s) < (1/5)(1 + c/2)(2 - s)` for some `s in [3/2, 2)`, `g(s) = log lambda(s,J)/log b`, `c = log(b-1)/log b`, met for `(9,0)` and `(9,8)` with a margin `7.53 x 10^(-3)`, and verbatim `For all other pairs (b,a_0) != (9,0) and (b,a_0) != (9,8), the above condition can't be fulfilled. Mainly because c = log(b-1)/log b decreases more strongly than g(s) does, if b decreases. Thus, a refinement of Maynard's estimates in [2] or a completely new approach is necessary to make progress for bases b <= 9`; the introduction records that the previous article closes every base `b >= 10`; the dissertation `Primzahlen mit einer ausgeschlossenen Ziffer` is its reference [1]; the only source is this PDF on the author's university page, no DOI and no arXiv id | https://www.math.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Mathematik/20_Institut-Lehrstuehle/5_Algebra_und_Zahlentheorie/Karwatowski/Digits_of_primes_in_base_b_9.pdf |
| Karwatowski 2022 | Primes with one excluded digit, Acta Arith. 202, 105-121, doi:10.4064/aa191002-26-8, resolved on Crossref; not read at source, carried from two restatements read at source: the author's base-9 paper (`the above conjecture is true for all b >= 10`) and Granville 2024, which states that Karwatowski used the largest row sum as an eigenvalue bound to prove numerically `lambda_(4,1) < q^(27/77)` and `lambda_(4,235/154) < q^(59/433)` for all `q >= 10`, Maynard having shown them for `q = 10` | https://doi.org/10.4064/aa191002-26-8 |
| Granville 2024 | Missing digits, and good approximations, Bull. Amer. Math. Soc. 61, write-up of the 2023 AMS Current Events Bulletin lecture, read at source in arXiv:2308.03126, 31 pages: the survey of Maynard's missing-digit proof and of Koukoulopoulos-Maynard on Duffin-Schaeffer; the sentence on Karwatowski quoted in the row above; the lectures name no result for a base below 10 | https://arxiv.org/abs/2308.03126 |
| van der Corput 1923 | Zahlentheoretische Abschaetzungen mit Anwendung auf Gitterpunktprobleme, Math. Z. 17, 250-259. Satz 5 bounds a sawtooth sum under an affine substitution, so an arithmetic progression is admissible at every fixed modulus and the shallow half of a circular arc is capped by an unconditional power saving with no modern exponent in it. Cited on [crop](notes/crop.md) | https://doi.org/10.1007/BF01504346 |
| Davenport, Multiplicative Number Theory, zero-free regions | Graduate Texts in Mathematics 74, Springer. Chapter 14 Zero-Free Regions for L(s, chi) carries the classical region and the effective Landau-Page bound on the exceptional real zero; chapter 20 The Prime Number Theorem for Arithmetic Progressions (I) carries the effective form; chapter 21 is Siegel's theorem, the ineffective step the base-smooth conductor bound removes the need for. This row is not the Davenport bound `x (log x)^(-A)` for `Sum mu(n) e(n theta)` cited elsewhere on `mobius.md`, which is carried at source in Porritt 2018. DOI resolved on Crossref. | https://doi.org/10.1007/978-1-4757-5927-3 |
| Laugesen and Liu 2016 | Optimal stretching for lattice points and eigenvalues, arXiv:1609.06172, Appendix A The van der Corput sum, Theorem 18: van der Corput's Satz 5 in explicit form, `abs(sum_(a<n<=b) psi(h(n))) <= 6 int abs(h'')^(1/3) + 175 max abs(h'')^(-1/2) + 1`, citing Satz 5 and Kraetzel's Korollar zu Satz 1.5 p. 24 and sharpening Kraetzel's `+2` to `+1`. Read at source; the `+2` is what [crop](notes/crop.md) uses. | https://arxiv.org/abs/1609.06172 |
| Beurling 1937 | Analyse de la loi asymptotique de la distribution des nombres premiers generalises. I, Acta Mathematica 68, 255-291 - the generalised prime systems; bibliographic record resolved on Crossref, Springer refuses fetchers | https://doi.org/10.1007/BF02546666 |
| Diamond, Montgomery and Vorhauer 2006 | Beurling primes with large oscillation, Mathematische Annalen 334(1), 1-36 - a Beurling system with integer count `N_B(x) = kappa x + O(x^theta)`, `1/2 < theta < 1`, whose zeta has infinitely many zeros on `sigma = 1 - a/log t` and none to the right, so RH can fail for a Beurling system with a regular integer count; record resolved on Crossref, abstract read at source, full text not | https://doi.org/10.1007/s00208-005-0638-2 |
| Turing 1952 | The chemical basis of morphogenesis, Philosophical Transactions B 237, 37-72; reaction-diffusion pattern selection by a linear instability at one wavelength | https://doi.org/10.1098/rstb.1952.0012 |
| Evans 2001 | Larger than Life: Digital Creatures in a Family of Two-Dimensional Cellular Automata, DMTCS Proceedings AA; Life rescaled to box neighbourhoods of range r with rescaled gliders and spaceships | https://doi.org/10.46298/dmtcs.2288 |
| Pivato 2007 | RealLife: the continuum limit of Larger Than Life cellular automata, Theoretical Computer Science 372; Larger than Life at large radius has life forms virtually identical after rescaling | https://arxiv.org/abs/math/0503504 |
| Rafler 2011 | Generalization of Conway's Game of Life to a continuous domain - SmoothLife; ring neighbourhood and disk cell, typically outer radius three times the inner | https://arxiv.org/abs/1111.1567 |
| Chan 2019 | Lenia - Biology of Artificial Life, Complex Systems 28(3); scale invariance through large neighbourhoods with exponential, polynomial and rectangular kernel cores | https://arxiv.org/abs/1812.05433 |
| Chladni 1787 | Entdeckungen ueber die Theorie des Klanges; the sand figures of vibrating plates, digitised by the Deutsches Textarchiv | https://www.deutschestextarchiv.de/book/show/chladni_klang_1787 |
| Davies, Gladwell, Leydold and Stadler 2001 | Discrete nodal domain theorems, Linear Algebra and its Applications 336, 51-60; an eigenvector of index k and multiplicity r on a connected graph has at most k + r - 1 strong and k weak nodal domains | https://arxiv.org/abs/math/0009120 |
| Morrison 2004 | Cosine Products, Fourier Transforms, and Random Sums; the infinite cosine product whose Fourier transform is the Cantor measure | https://arxiv.org/abs/math/0411380 |
| Tsujii 2012 | On the Fourier transforms of self-similar measures; the transform of a self-similar measure as an infinite product | https://arxiv.org/abs/1212.1553 |
| Liu, Wang and Zheng 2026 | Spectrality of product-form self-similar measures and tiles; the digit-set transform as the product of the mask polynomial at scaled frequencies | https://arxiv.org/abs/2603.21031 |
| Sapoval, Gobron and Margolina 1991 | Vibrations of Fractal Drums, Physical Review Letters 67, 2974; localized low-frequency modes of a membrane with a fractal boundary | https://doi.org/10.1103/PhysRevLett.67.2974 |
| Baake 1997 | Solution of the coincidence problem in dimensions d <= 4, in The Mathematics of Long-Range Aperiodic Order, Kluwer, 9-44; read at source in arXiv:math/0605222, Prop. 3.1 giving the coincidence-site-lattice Dirichlet series of the square lattice `(1 + 2^-s)^-1 zeta_Q(i)(s)/zeta(2s)` | https://arxiv.org/abs/math/0605222 |
| Baez-Duarte 2003 | A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Rend. Lincei Mat. Appl. 14, 5-11; read at source in arXiv:math/0202141; the criterion as a stack of coarsening sawtooths with integer indices | https://arxiv.org/abs/math/0202141 |
| Beurling 1955 | A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. USA 41(5), 312-314; record read at source, page images only | https://doi.org/10.1073/pnas.41.5.312 |
| Davenport 1937 | On some infinite series involving arithmetical functions, Quart. J. Math. Oxford 8, 8-13; the article is paywalled and its identities are read in the renderings arXiv:2005.08279 eq. 1.1, arXiv:1205.2343 and arXiv:1708.00115 | https://academic.oup.com/qjmath/article-abstract/os-8/1/8/1564852 |
| Patkowski 2017 | On arithmetic series involving the fractional part function, arXiv:1708.00115; eq. 1.1 is Davenport's von Mangoldt identity verbatim | https://arxiv.org/abs/1708.00115 |
| Sayous 2024 | Gaps in the complex Farey sequence of an imaginary quadratic number field, arXiv:2407.04380; the complex Farey set, its equidistribution with no rate and its `zeta_K(2)` count, with no mention of Franel, Landau or RH | https://arxiv.org/abs/2407.04380 |
| Estala-Arias 2019 | Discrete measures and the extended Riemann hypothesis, arXiv:1908.03658; RH for `zeta_K` as a convergence rate of measures on the positive reals built from ideal norms, not on `C/O_K` | https://arxiv.org/abs/1908.03658 |
| Smith 1875 | On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7, 208-212; `det(gcd(i,j)) = prod phi(k)` and the general `prod (f * mu)(k)`, statement read in two independent renderings, the article behind a paywall | https://doi.org/10.1112/plms/s1-7.1.208 |
| Gilbert 1986 | The fractal dimension of sets derived from complex bases, Canad. Math. Bull. 29, 495-500; abstract read at source: the radix representation of the complex numbers in the base `-n + i` tiles the plane | https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/fractal-dimension-of-sets-derived-from-complex-bases/1E21659EEEECF5215C7486531ACCA9AF |
| Jaffard 2012 | Multivariate Davenport series, arXiv:1205.2343; the introduction carries Davenport's expansion `sum a_n {nx} = sum_m c_m sin(2 pi m x)` with `c_m = -(1/(pi m)) sum over n dividing m of n a_n` | https://arxiv.org/abs/1205.2343 |
| Huxley 1971 | The distribution of Farey points I, Acta Arith. 18, 281-287; the theorem on p. 282 with the character equivalence, Lemma (10) on p. 283 giving the Fourier `L^2` Franel identity over the rationals, and the p. 283 announcement of a Farey sequence for an algebraic number field; read at source in the archived scan, the live journal page refusing every client | https://web.archive.org/web/2020id_/http://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa18130.pdf |
| Huxley 2012 | Identities involving Farey fractions, Trudy Mat. Inst. Steklova 276, 131-145; Theorem 1 on p. 133 the weighted Fourier Franel identity over the rationals, and p. 134 reporting the planned number-field part II abandoned | https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=3374&option_lang=eng |
| Cobeli and Zaharescu 2003 | The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis 5, 1-38; the survey's complex-plane generalisations are Diophantine geometry and its RH equivalences are over the rationals only | https://emis.dsd.sztaki.hu/journals/AUA/acta5/survey3.ps_pages1-20.pdf |
| Kanemitsu and Yoshimoto 1996 | Farey series and the Riemann hypothesis, Acta Arith. 75; Theorem 3 prints Franel's formula and generalises the weight to Kubert functions and the range to short intervals, with no number-field version | https://eudml.org/doc/206882 |
| Weyl 1916 | Uber die Gleichverteilung von Zahlen mod. Eins, Mathematische Annalen 77, 313-352; the equidistribution of `n^2 delta` mod 1 at irrational `delta`; full scan | https://eudml.org/doc/158730 |
| Murty and Pathak | Evaluation of the quadratic Gauss sum; Theorem 1.1 gives `sqrt n`, `0`, `i sqrt n`, `(1 + i) sqrt n` by `n mod 4`; read at source | https://www.cmi.ac.in/~siddhi/Quadratic_Gauss_Sum.pdf |
| Hedenmalm, Lindqvist and Seip 1997 | A Hilbert space of Dirichlet series and systems of dilated functions in `L^2(0,1)`, Duke Math. J. 86, 1-37; read at source in arXiv:math/9512211; the Riesz-basis criterion for dilation systems and the completeness and minimality results the stack's layers are read against | https://arxiv.org/abs/math/9512211 |
| Lewko and Radziwill 2014 | Refinements of Gal's theorem and applications, arXiv:1408.2334; the spectral-norm form of Gal's bound on gcd matrices | https://arxiv.org/abs/1408.2334 |
| Pleasants, Baake and Roth 1996 | Planar coincidences for N-fold symmetry, J. Math. Phys. 37, 1029-1058; read in the arXiv source; section 4, the triangular lattice example after the theorem closing section 3, gives the triangular lattice's coincidence Dirichlet series `(1 + 3^-s)^-1 zeta_K(s)/zeta(2s)` and the average number of coincidence lattices `sqrt 3/(2 pi)` | https://arxiv.org/abs/math/0511147 |
| Hu, Kaneko, Martin and Schildkraut 2023 | On a Mertens-type conjecture for number fields, Math. Proc. Cambridge Philos. Soc.; read at source in arXiv:2109.06665; the Mellin identity 1.1 for `M_K`, Lemma 5.4 bounding `log zeta_K` right of the critical line under the hypothesis, and Lemma 2.4 the good horizontal lines | https://arxiv.org/abs/2109.06665 |
| Wieleitner 1908 | Spezielle ebene Kurven, section 25 Die trochoidalen Kurven, numbers 164 to 190, the trochoidal material proper being 164 to 175 | https://archive.org/details/spezielleebeneku00wielrich |
| Sakhnovich 2023 | Periodic functions: self-intersections, local singular points, and folds | https://arxiv.org/abs/2304.09940 |
| Dresden 2025 | Epitrochoids and Hypotrochoids Together Again, The College Mathematics Journal 56(3) 203-213 | https://doi.org/10.1080/07468342.2024.2388009 |
| Hunter 1957 | The minimum discriminants of quintic fields | https://doi.org/10.1017/S2040618500033463 |
| Pohst 1982 | On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields | https://doi.org/10.1016/0022-314X(82)90061-0 |
| Jones and Roberts 2014 | A database of number fields | https://arxiv.org/abs/1404.0266 |
| Hutchinson 1981 | Fractals and Self Similarity, Indiana Univ. Math. J. 30(5), 713-747 - read at source in the author's retypeset copy: 5.1(2) the unique D with sum r_i^D = 1, 5.1(3) names it the similarity dimension, 5.1(4)(i) dim K <= D for arbitrary contractions, 5.2(1) the open set condition, 5.3(1) 0 < H^D(K) < infinity and dim K = D under it; 3.3(2) the Koch curve as the attractor of four similitudes, the ratio and the angles in Figure 3.2 only, not in the text | https://doi.org/10.1512/iumj.1981.30.30055 |
| Lind and Marcus 1995 | An Introduction to Symbolic Dynamics and Coding, Cambridge University Press - chapters 1 to 3 read at source, chapter 4 not opened: Theorem 2.3.2, an M-step shift of finite type X has a graph G with X^[M+1] = X_G; Proposition 2.3.9(3), X^[M] is a 1-step shift of finite type; the entropy of an irreducible shift of finite type as the logarithm of the Perron eigenvalue of its adjacency matrix is carried here with no theorem number | https://doi.org/10.1017/CBO9780511626302 |
| Lagarias and Wang 1997 | Integral self-affine tiles in R^n II. Lattice tilings, J. Fourier Anal. Appl. 3, 83-102 - review only, the statement read in the verbatim restatement of Steiner and Thuswaldner, Rational self-affine tiles, arXiv:1203.0758, citing LW97 Corollary 6.2 and Lemma 2.1: for an expanding integer matrix A with irreducible characteristic polynomial and D a complete set of coset representatives of Z^n / A Z^n, the tile T(A, D) induces a lattice tiling of R^n with respect to the smallest A-invariant sublattice containing D - D; the irreducibility hypothesis excludes a rational integer base | https://doi.org/10.1007/BF02649100 |
| Patterson 1944 | Ambiguities in the X-ray analysis of crystal structures, Phys. Rev. 65, 195-201 | https://doi.org/10.1103/PhysRev.65.195 |
| Rosenblatt and Seymour 1982 | The structure of homometric sets, SIAM J. Algebraic Discrete Methods 3, 343-350 | https://doi.org/10.1137/0603035 |
| Kac 1966 | Can one hear the shape of a drum?, Amer. Math. Monthly 73, no. 4, 1-23 | https://doi.org/10.2307/2313748 |
| Gordon, Webb and Wolpert 1992 | One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27, 134-138 | https://doi.org/10.1090/S0273-0979-1992-00289-6 |
| Hu 1962 | Visual pattern recognition by moment invariants, IRE Trans. Inform. Theory 8, 179-187 | https://doi.org/10.1109/TIT.1962.1057692 |
| Teague 1980 | Image analysis via the general theory of moments, J. Opt. Soc. Amer. 70, 920-930 | https://doi.org/10.1364/JOSA.70.000920 |
| Kontorovich and Oh 2011 | Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, with an appendix by Hee Oh and Nimish Shah, J. Amer. Math. Soc. 24, 603-648; the number of circles of curvature at most T in a bounded Apollonian packing is asymptotic to c T^alpha with alpha the residual dimension, alpha ~ 1.30568(8), read at source in the arXiv v5 | https://arxiv.org/abs/0811.2236 |
| Vytnova and Wormell 2024 | Hausdorff dimension of the Apollonian gasket, arXiv:2406.04922; Theorem 1.1 gives dim_H = 1.3056867280 4987718464 5986206851 0408911060 ... +- 10^(-129), rigorous to 128 places, and records the earlier readings | https://arxiv.org/abs/2406.04922 |
| McMullen 1998 | Hausdorff dimension and conformal dynamics III: Computation of dimension, Amer. J. Math. 120, 691-721; the eigenvalue algorithm and H. dim of the Apollonian gasket approximately 1.305688 | https://people.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf |
| Bai and Finch 2018 | Precise calculation of Hausdorff dimension of Apollonian gasket, Fractals 26(4), 1850050; an induced transfer operator on the complex unit circle reaching 30 places non-rigorously, paywalled and read only through Vytnova and Wormell 2024 | https://doi.org/10.1142/S0218348X18500500 |
| Haag, Kertzer, Rickards and Stange 2023 | The local-global conjecture for Apollonian circle packings is false, arXiv:2307.02749, Annals of Mathematics 200(2); quadratic and quartic families are missed, the obstruction a property of the thin Apollonian group and not its Zariski closure, from quadratic and quartic reciprocity | https://arxiv.org/abs/2307.02749 |
| Boyd 1973 | The residual set dimension of the Apollonian packing, Mathematika 20(2), 170-174; the bracket 1.300197 < d_H < 1.314534, the first rigorous one, read only through Vytnova and Wormell 2024 | https://doi.org/10.1112/S0025579300004745 |
| Kari 2012 | Cellular Automata, the Collatz Conjecture and Powers of 3/2, in Developments in Language Theory (DLT 2012), Lecture Notes in Computer Science 7410, 40-49. Metadata verified at source through the Crossref record of the DOI: author Kari, pages 40-49, year 2012. Abstract unverified: the chapter page redirects to the publisher login and the Crossref record carries no abstract, so no sentence of the chapter is quoted or restated on the page, which cites the title alone | https://doi.org/10.1007/978-3-642-31653-1_5 |
| Bruschi 2005 | Two Cellular Automata for the 3x+1 Map, arXiv:nlin/0502061. Abstract read at source: "Two simple Cellular Automata, which mimic the Collatz-Ulam iterated map (3x+1 map), are introduced. These Cellular Automata allow to test efficiently the Collatz conjecture for very large numbers." Abstract only | https://arxiv.org/abs/nlin/0502061 |
| Chen 2013 | Cellular Automata to More Efficiently Compute the Collatz Map, arXiv:1301.3125. Abstract read at source: "The Collatz, or 3x+1, Conjecture claims that for every positive integer n, there exists some k such that T^k(n)=1, where T is the Collatz map. We present three cellular automata (CA) that transform the global problem of mimicking the Collatz map in bases 2, 3, and 4 into a local one of transforming the digits of iterates." Abstract only; the bases are 2, 3 and 4, not 6 | https://arxiv.org/abs/1301.3125 |
| Cobham 1969 | On the base-dependence of sets of numbers recognizable by finite automata, Math. Systems Theory 3, 186-192; the law that base-locks the whole one-base method, stated and quoted through Bes Theorem 24 and Durand and Rigo Theorem 1.1, both read at source, the original being paywalled | https://doi.org/10.1007/BF01746527 |
| Semenov 1977 | Presburgerness of predicates regular in two number systems, Siberian Math. J. 18, 289-300; the extension of Cobham to subsets of N^n with Presburger definability in place of ultimate periodicity, quoted through Bes Theorem 25 and Durand and Rigo Theorem 4.7, both read at source, the original being paywalled | https://doi.org/10.1007/BF00967164 |
| Durand and Rigo | On Cobham's theorem, a survey chapter; Definition 1.1 for multiplicative independence, Remark 1.2 for the dependence classes, Theorem 1.1 for Cobham and Theorem 4.7 for Cobham-Semenov | https://orbi.uliege.be/bitstream/2268/39461/1/Chapter26.pdf |
| Bes | A Survey of Arithmetical Definability; Theorem 4 for semilinear equals Presburger-definable after Ginsburg and Spanier, Theorem 20 for the Buchi characterization, Theorem 24 for Cobham and Theorem 25 for Cobham-Semenov, and the three-class split of the subsets of N | https://lacl.u-pec.fr/bes/publi/survey.pdf |
| Furstenberg 1970 | Intersections of Cantor Sets and Transversality of Semigroups, in Problems in Analysis, Princeton University Press, 41-60; the source Shmerkin 2019 names for the intersection conjecture | https://doi.org/10.1515/9781400869312-004 |
| Corso and Shmerkin 2024 | Dynamical self-similarity, L^q-dimensions and Furstenberg slicing in R^d, arXiv:2409.04608; Theorem 1.15 and Corollary 1.17 read at source give the m-fold intersection bound for pairwise multiplicatively independent bases, one closed invariant subset of the circle per base | https://arxiv.org/abs/2409.04608 |
| Yu 2021b | An improvement on Furstenberg's intersection problem, accepted in Trans. Amer. Math. Soc.; the m >= 3 bound under a Q-linear independence hypothesis on the log ratios, superseded unconditionally by Corso and Shmerkin 2024 | https://arxiv.org/abs/1811.11073 |
| Glasscock, Moreira and Richter 2024 | Additive and geometric transversality of fractal sets in the integers, J. London Math. Soc. 109, e12902; the integer analogues of the Furstenberg transversality conjectures for xr- and xs-invariant sets in multiplicatively independent bases | https://arxiv.org/abs/2007.05480 |
| Burrell and Yu 2021 | Digit expansions of numbers in different bases, arXiv:1905.00832v3; Theorem 1.2 and Theorem 1.4 for the base 4 and 5 count, Theorem 1.6 under Schanuel's conjecture, Theorem 1.11 on Q-linear independence, and Theorem 1.8 quoting Erdos, Graham, Ruzsa and Straus | https://arxiv.org/abs/1905.00832 |
| Senge and Straus 1973 | PV-numbers and sets of multiplicity, Period. Math. Hungar. 3, 93-100; finiteness of the integers with bounded digit sums in two bases exactly when the bases are multiplicatively independent, ineffective through Thue-Siegel-Roth, read through Bugeaud, Cipu and Mignotte, the original being paywalled | https://doi.org/10.1007/BF02018464 |
| Stewart 1980 | On the representation of an integer in two different bases, J. reine angew. Math. 319, 63-72; the effective form of Senge and Straus through Baker's linear forms in logarithms, read through Bugeaud, Cipu and Mignotte, the original being paywalled | https://doi.org/10.1515/crll.1980.319.63 |
| Erdos, Graham, Ruzsa and Straus 1975 | On the prime factors of C(2n, n), Math. Comp. 29, 83-92; the only lower bound in the two-base digit literature, quoted verbatim as Theorem 1.8 of Burrell and Yu, the original not opened | https://doi.org/10.1090/S0025-5718-1975-0369288-3 |
| Bugeaud, Cipu and Mignotte | On the representation of Fibonacci and Lucas numbers in an integer base; the survey that carries the Senge and Straus statement and Stewart's effective inequality verbatim | https://irma.math.unistra.fr/~bugeaud/travaux/Ribfinal1.pdf |
| Collatz 1942 | Einschliessungssatz fuer die charakteristischen Zahlen von Matrizen, Mathematische Zeitschrift 48(1), 221-226; the first half of the Collatz-Wielandt name, the min-max characterization of the Perron root and not the 3n + 1 map. The original is paywalled at Springer; the statement is read through the Wikipedia Perron-Frobenius article and the metadata verified in the Crossref record of the DOI, author Collatz, volume 48, pages 221-226, year 1942 | https://doi.org/10.1007/BF01180013 |
| Wielandt 1950 | Unzerlegbare, nicht negative Matrizen, Mathematische Zeitschrift 52(1), 642-648; the second half of the Collatz-Wielandt name, the irreducible case behind the brackets this tree certifies with. The original is paywalled at Springer; the statement is read through the Wikipedia Perron-Frobenius article and the metadata verified in the Crossref record of the DOI, author Wielandt, volume 52, pages 642-648, year 1950 | https://doi.org/10.1007/BF02230720 |
| divisor avatars | Divisor Avatars: Which Parity Designs Count the Divisors of a Power | https://github.com/carlomitchener/carlomitchener/tree/main/research/divisor-avatars |

## REFERENCE PAGES

| ref | title | url |
|---|---|---|
| Menger sponge article | Wikipedia: the centroid cross-section perpendicular to a space diagonal, hexagram recurrence a_n = 9a_(n-1) - 12a_(n-2), cross-referenced to A299916. It sources the hexagrams to a newspaper piece and the count to A299916 itself, so it is not independent of the OEIS comment | https://en.wikipedia.org/wiki/Menger_sponge |
| n-flake article | Wikipedia: octahedron flake of dimension log(6)/log(2), and the Cantor cube projecting to a hexaflake | https://en.wikipedia.org/wiki/N-flake |
| Eppstein, Geometry Junkyard | Sierpinski Tetrahedra and Other Fractal Sponges - four equivalent constructions, one being Pascal's Pyramid mod 2 | https://ics.uci.edu/~eppstein/junkyard/sierpinski.html |
| Kummer's theorem | Wikipedia: the 2-adic valuation of C(i+j,i) counts base-2 carries, so C(i+j,i) is odd iff i AND j = 0 | https://en.wikipedia.org/wiki/Kummer%27s_theorem |
| Burnside's lemma | Wikipedia: the orbit-counting average the bijection and base-q census pages run on | https://en.wikipedia.org/wiki/Burnside%27s_lemma |
| Franel-Landau theorem | Wikipedia, Farey sequence: the RH-equivalent discrepancy statements | https://en.wikipedia.org/wiki/Farey_sequence |
| Landau-Ramanujan constant | Wikipedia: the count of sums of two squares below X is asymptotic to K X / sqrt(ln X), Landau 1908 | https://en.wikipedia.org/wiki/Landau%E2%80%93Ramanujan_constant |
| Redheffer matrix | Wikipedia: the 0-1 divisibility-incidence matrix with det A_n = M(n), named and killed as a steelman on [farey](notes/farey.md) | https://en.wikipedia.org/wiki/Redheffer_matrix |
| Ostrowski numeration | Wikipedia: the continued-fraction positional system behind the log-time floor-sum counts that keep irrational points computable on [farey](notes/farey.md) | https://en.wikipedia.org/wiki/Ostrowski_numeration |
| Jacobi two-square theorem | Wikipedia, Sum of squares function: r_2(n) = 4*(d_1(n) - d_3(n)), whose Dirichlet series is 4*zeta(s)*beta(s) | https://en.wikipedia.org/wiki/Sum_of_squares_function |
| Perron-Frobenius theorem | Wikipedia: the Collatz-Wielandt formula, the min-max over positive vectors behind the even-half certificates, now the `slice-sign-even-half` lane | https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem |
| Metzler matrix | Wikipedia: nonnegative off-diagonal entries; `M_even - (fill/3) I` is one, which is why the M-matrix classification of the shifted block is circular | https://en.wikipedia.org/wiki/Metzler_matrix |
| NKS note, rule 150 | A New Kind of Science note for page 885: there are 2^m Fibonacci[m+2] black cells up to step 2^m, so the fractal dimension is Log[2, 1 + Sqrt[5]] | https://www.wolframscience.com/nks/notes-3-2--rule-150/ |
| NKS note, additive dimensions | A New Kind of Science note for page 955: Log[2,3] for rule 90 and Log[2, 1+Sqrt[5]] for rule 150, with the general recipe. Not a refereed proof; the theorem is Willson's | https://www.wolframscience.com/nks/notes-6-6--fractal-dimensions-of-additive-cellular-automata/ |
| NKS note, surjectivity | A New Kind of Science note for page 959: 30 surjective elementary rules, and in two dimensions such properties are in general undecidable | https://www.wolframscience.com/nks/notes-6-7--surjectivity-and-injectivity-of-cellular-automaton-maps/ |
| NKS page 436 | A New Kind of Science: of the 256 elementary rules only six are reversible; the page does not name them | https://www.wolframscience.com/nks/p436--the-notion-of-reversibility/ |
| DLMF 25.13 | Periodic Zeta Function - 25.13.1 defines `F(x,s) = sum_(n>=1) e^(2 pi i n x)/n^s`, convergent for `Re s > 0` off the integers, and 25.13.3 is Hurwitz's formula `zeta(1-s,x) = (Gamma(s)/(2 pi)^s)(e^(-pi i s/2) F(x,s) + e^(pi i s/2) F(-x,s))` for `Re s > 0`, `0 < x < 1`; read at source, both quoted verbatim. The same page carries 25.13.2, which is the formula the reflection step recovers with the range `Re s > 0`. This is the kernel of the position-product identity | https://dlmf.nist.gov/25.13 |
| DLMF 25.12 | Polylogarithms - 25.12.12 gives `Li_s(z) = Gamma(1-s)(log(1/z))^(s-1) + sum_(n>=0) zeta(s-n)(log z)^n/n!` for `s` not a positive integer and `abs(log z) < 2 pi`; read at source. Folding `t > 1/2` to `1-t` holds the argument at `abs(mu) <= pi`, so the series is geometric at ratio `1/2` and gives the fast evaluator used for the periodic zeta | https://dlmf.nist.gov/25.12 |
| Niven's theorem | the only rational cosines at rational multiples of `pi` are `0, +-1/2, +-1`; Niven, Irrational Numbers, Carus 11, Cor. 3.12; the statement also read in the abstract of arXiv:2508.06415 | https://mathworld.wolfram.com/NivensTheorem.html |
| Erb rhodonea | Rhodonea curves | https://www.math.unipd.it/~erb/rhodonea.html |
| MathWorld hypotrochoid | Hypotrochoid | https://mathworld.wolfram.com/Hypotrochoid.html |
| MathWorld epitrochoid | Epitrochoid | https://mathworld.wolfram.com/Epitrochoid.html |
| MathWorld rose | Rose Curve | https://mathworld.wolfram.com/RoseCurve.html |
| Jaekel hypotrochoids | Complete investigation of the shape diversity of Hypotrochoids / Hypocycloids | https://www.v-jaekel.de/hypo-h/formenvielfalt-einer-hypotrochoide-en.html |
| OEIS A000616 | a(-1)=1 by convention; for n >= 0, a(n) = number of irreducible Boolean functions of n variables. | https://oeis.org/A000616 |
| OEIS A191363 | Numbers m whose deficiency is 2: sigma(m) - 2*m = -2. | https://oeis.org/A191363 |
| OEIS A000045 | Fibonacci numbers | https://oeis.org/A000045 |
| OEIS A000930 | Narayana's cows sequence | https://oeis.org/A000930 |
| Ramanujan's sum | Ramanujan's sum, Wikipedia: the divisor form c_q(n) = sum over d dividing gcd(q,n) of mu(q/d) d, published by Kluyver in 1906, the sums named after Ramanujan's 1918 paper; the article cites Hardy and Wright Theorems 65 and 66 for the root-of-unity facts and NOT Theorem 272, so the Theorem 272 number often quoted for Ramanujan's sum is unconfirmed and must not be printed; read at source | https://en.wikipedia.org/wiki/Ramanujan%27s_sum |
| A001146 | a(n) = 2^(2^n). | https://oeis.org/A001146 |
| A001622 | Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2. | https://oeis.org/A001622 |
| A007582 | a(n) = 2^(n-1)*(1+2^n). | https://oeis.org/A007582 |
| A058265 | Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1. | https://oeis.org/A058265 |
| A060006 | Decimal expansion of real root of x^3 - x - 1 (the plastic constant). | https://oeis.org/A060006 |
| A092526 | Decimal expansion of (2/3)*cos( (1/3)*arccos(29/2) ) + 1/3, the real root of x^3 - x^2 - 1. | https://oeis.org/A092526 |
| A003714 | Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's. | https://oeis.org/A003714 |
| A000225 | a(n) = 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.) | https://oeis.org/A000225 |
| A030979 | Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7; the Kummer neighbour of the three-base thin set, one base-7 digit wider, carrying a prize and a heuristic of Pomerance whose exponent is the transversality budget 0.025951, and a table its entry calls complete to 10^70 with 1374 terms | https://oeis.org/A030979 |
| A001792 | a(n) = (n+2)*2^(n-1); the level-1 cell count of the Menger sponge in every dimension | https://oeis.org/A001792 |
| A005408 | The odd numbers; the odd-side fills of the dimension-one solid | https://oeis.org/A005408 |

## PRIOR ART ON THE BASE-3 SLICE

The upstream that [spectra](notes/spectra.md) reads its base-3 rung against. It is grey literature: a photograph, a video, three blog posts and an OEIS comment, with two peer-reviewed generalisations that move along dimension or change the solid, never along base.

| ref | title | url |
|---|---|---|
| Perez-Duarte, "Slice of Menger" | Flickr: the base-3 centroid diagonal cut, "a very interesting pattern of stars and hexagons"; Abel credits it as first. The companion animated cross-section at `1438621219` now returns 404 | https://www.flickr.com/photos/sbprzd/1432723128/ |
| Hart, "Mathematical Impressions" | Simons Foundation: The Surprising Menger Sponge Slice, the video that popularised the cut | https://www.simonsfoundation.org/2012/12/10/mathematical-impressions-the-surprising-menger-sponge-slice/ |
| Hart, mirrored | the same video at Scientific American; cite the Simons original | https://www.scientificamerican.com/article/mathematical-impressions-the-surprising-menger-sponge-slice/ |
| Cook 2011 | Code to slice open a Menger sponge - working Python from the base-3 digit predicate, point-sampled into a raster; its prose says the normal runs to `(1, 1, 1)` while the listing sets `normal = (1, 1, 0.5)` | https://www.johndcook.com/blog/2011/08/30/slice-a-menger-sponge/ |
| Abel, "Seeing Stars" | "replace each hexagon with 6 hexagons and 6 triangles, and replace each triangle with 1 hexagon and 3 triangles", and `d = log_3((9+sqrt(33))/2) = 1.8184`, with the author's own hedge that it is a computation and not yet a full proof. The substitution is reproved by exhaustion in [README](README.md) | http://blog.zacharyabel.com/2012/02/seeing-stars/ |
| Abel, "A Slice of Interdimensional Sponge Cake" | the same dimension stated verbatim; "Seeing Stars" derives it | http://blog.zacharyabel.com/2012/02/a-slice-of-interdimensional-sponge-cake/ |
| Chang, "The Mystery of the Menger Sponge" | New York Times: quotes Hart on the diagonal slice showing six-sided stars, as a proposed exhibit; the newspaper piece Wikipedia cites beside the OEIS comment. Live URL paywalled to fetchers | https://www.nytimes.com/2011/06/28/science/28math-menger.html |
| Hocking, Bridges 2023 | Three-Dimensional Diagonal Cross-Sections of Four-Dimensional Menger Sponges - generalises the cut along dimension `n` and along a hole-iness parameter `k`, with "the base three expansion" fixed throughout | https://archive.bridgesmathart.org/2023/bridges2023-291.pdf |
| Hocking, Bridges 2024 | Menger-Slice Inspired Fractals based on the Pentagon, Dodecahedron, and 120-Cell - frames the Menger slice as a two-tile closed fractal family and writes "The literature uses the term 'directed-graph iterated function system'", which is [spectra](notes/spectra.md)'s own structural claim at base 3. Different solid, same method: it owns the grammar move without touching the base generalisation | https://archive.bridgesmathart.org/2024/bridges2024-297.pdf |

## DATASETS

| ref | title | url |
|---|---|---|
| Bourke page | Paul Bourke, mrly fractals: Menger, Sierpinski, Cantor - states 44/81 for sponge5 and 135/208 for sponge7 | https://paulbourke.net/fractals/mrlymath/ |
| Source PDF | Marley Math: Cantor Sets, Sierpinski Carpets, Menger Sponges, And More - the published rendering of these families | https://paulbourke.net/fractals/mrlymath/mrlymath.pdf |
| b001316 | OEIS b-file for A001316, 50001 terms | https://oeis.org/A001316/b001316.txt |
| b047999 | OEIS b-file for A047999, 10585 terms (rows 0..144) | https://oeis.org/A047999/b047999.txt |
| OEIS dump | `stripped.gz`, the full-sequence dump the novelty searches on [sequences](sequences.md) run against; 398556 lines | https://oeis.org/stripped.gz |
| OEIS submission rules | Submit.html, the Style Sheet and the AI-submission policy every draft in [sequences](sequences.md) is written against | https://oeis.org/wiki/Style_Sheet |
| LMFDB 2.0.3.1 | LMFDB number field 2.0.3.1 | https://www.lmfdb.org/NumberField/2.0.3.1 |
| LMFDB 2.2.5.1 | LMFDB number field 2.2.5.1 | https://www.lmfdb.org/NumberField/2.2.5.1 |
| LMFDB 3.1.23.1 | LMFDB number field 3.1.23.1 | https://www.lmfdb.org/NumberField/3.1.23.1 |
| LMFDB 3.3.49.1 | LMFDB number field 3.3.49.1 | https://www.lmfdb.org/NumberField/3.3.49.1 |
| LMFDB 4.0.117.1 | LMFDB number field 4.0.117.1 | https://www.lmfdb.org/NumberField/4.0.117.1 |
| LMFDB 4.0.1225.1 | LMFDB number field 4.0.1225.1 | https://www.lmfdb.org/NumberField/4.0.1225.1 |
| LMFDB 4.4.725.1 | LMFDB number field 4.4.725.1 | https://www.lmfdb.org/NumberField/4.4.725.1 |
| LMFDB 5.1.4429.1 | LMFDB number field 5.1.4429.1 | https://www.lmfdb.org/NumberField/5.1.4429.1 |
| LMFDB 5.3.4511.1 | LMFDB number field 5.3.4511.1 | https://www.lmfdb.org/NumberField/5.3.4511.1 |

## UNRESOLVED

Named on a page of this tree, with no citable source behind the name.

- Lorenz and Hardy, named in [bases](notes/bases.md) for the lattice-sum identity behind `4*zeta(s)*beta(s)`. The identity is normally traced to Lorenz 1871 and Hardy 1920 without a standard reference; the Wikipedia row above carries Jacobi's `r_2(n) = 4*(d_1(n) - d_3(n))`, of which the factorisation is the immediate Dirichlet transform.
- Dirichlet and Mertens, named in [pi](notes/pi.md) for the `6/pi^2` coprimality density - classical, with no paper named in this tree.
- The Chebyshev `psi` asymptotic, named in [coprime](notes/coprime.md) - classical, with no paper named in this tree.
- The Einstein relation `d_s = 2*d_f/d_w`, imported in [walks](notes/walks.md) - named but uncited, and the page already says it is a definition outside the classical cases.
- The accepted carpet numerics `d_w ~ 2.10` and `d_s ~ 1.80` in [walks](notes/walks.md) - attributed to stated literature values with no paper. Barlow and Bass 1999 is the rigorous foundation but does not itself state those decimals. This is the one unresolved name under a load-bearing claim.
- Zucker 1974 is cited in [bases](notes/bases.md) without a title. The row above is the paper that matches the parity-restricted lattice-sum claim, and the match is inferred, not stated on the page.
- Glaisher 1899's HathiTrust scan sits behind an automated challenge that blocks every automated check. The identical bibliographic record and the attribution both appear on OEIS A001316, which is the independent confirmation.
- `blog.zacharyabel.com` serves https under a `*.scripts.mit.edu` certificate and 503s intermittently over http, so both Abel rows can need a web archive snapshot.
- Three rows on the magic page have no open full text. Feng, Wen and Wu 1997 is the one with no reachable copy at all, so its theorem statements are carried from restatements; Moran 1946 gives bibliography and a first-page extract on Cambridge Core, and Cristea and Steinsky 2010 is read through its authors' own companion paper. Each row says so in place.
- A398348's own crossrefs, and the extension credits on A396934, are the OEIS's own attributions, read on the live entries.