Level tilings

• 2026-10-03 [Proved] For a digit set F at base b, Phi_m divides the level mask A_n(x) = prod_{i<n} F(x^(b^i)) if and only if m/gcd(m, b^i) lies in Z(F) = {m : Phi_m divides F} for some i < n, and the multiplicity of Phi_m in A_n is the sum over i < n of the multiplicity of Phi_{m/gcd(m, b^i)} in F. Witness: notes/tilings.md "The index lemma".
• 2026-10-03 [Verified] The cyclotomic part of 752 level masks, every digit set with 0 at bases 2 to 8, levels 1 to 3 and level 4 at bases up to 4, factored by PARI, equals the index lemma with multiplicity on all 752. Witness: lab/py/level-tiles index.
• 2026-10-03 [Verified] The Cantor level {0, 2, 6, 8} of {0, 2} at base 3 has mask Phi_4^2 Phi_12, reads T1 as 4 against 2, and has no complement in Z/N for multiples N of 4 up to 128. Witness: lab/py/level-tiles hand.
• 2026-10-03 [Proved] The tiling levels of a dimension-one design form an initial segment: if A_n tiles the integers then A_m tiles for every m <= n, so a digit set that does not tile the integers has no tiling level. Witness: notes/tilings.md "The tiling levels form an initial segment".
• 2026-10-03 [Proved] If F tiles Z/b then every level A_n tiles Z/b^n and hence the integers. Witness: notes/tilings.md "A digit set that tiles its residues tiles every level".
• 2026-10-03 [Proved] T1 holds at every level of F at base b if and only if F satisfies T1, Z_p(F) is empty for every prime p not dividing b, and the elements of Z_p(F) are pairwise incongruent mod v_p(b) for every p dividing b. Witness: notes/tilings.md "T1 at every level".
• 2026-10-03 [Proved] If every level of F at base b tiles the integers then #F divides b. Witness: notes/tilings.md "T1 at every level".
• 2026-10-03 [Proved] The T1 depth of F at base b is 0 if F fails T1, 1 if some prime off the base has Z_p(F) nonempty, and otherwise the least (c' - c)/v_p(b) over congruent pairs c < c' in some Z_p(F), infinite if there is none; for #F a prime power the tiling depth equals the T1 depth. Witness: notes/tilings.md "T1 at every level".
• 2026-10-03 [Proved] At a prime-power base every level of F tiles the integers if and only if F tiles Z/b. Witness: notes/tilings.md "T1 at every level".
• 2026-10-03 [Verified] {0, 1, 4, 5} at base 6 has tiling depth 2 and {0, 1, 8, 9} at base 10 has tiling depth 3, and the T1 depth law, the prime-power-size law and the prime-power-base law hold on all 65519 digit sets with 0 at bases 2 to 16. Witness: lab/py/level-tiles census, witness.
• 2026-10-03 [Refuted] Every level of F tiles the integers exactly when F tiles Z/b: {0, 2} at base 6 tiles at every level and does not tile Z/6. Witness: notes/tilings.md "The guess is false", lab/py/level-tiles witness.
• 2026-10-03 [Refuted] Every level of F tiles the integers exactly when F divided by its gcd tiles Z/b: {0, 1, 8, 9} at base 12, mask Phi_2 Phi_16, tiles at every level, and no integer multiple vF tiles Z/12, since Z_2(vF) = {1 + v_2(v), 4 + v_2(v)} and 2^(4 + v_2(v)) never divides 12, with an exact-cover search over v = 1..12 finding no complement. Witness: notes/tilings.md "The guess is false", lab/py/level-tiles witness.
• 2026-10-03 [Verified] Of the 65519 digit sets with 0 and at least two digits at bases 2 to 16, 588 tile Z/b, 609 tile at every level and 641 satisfy T1 at every level; the only every-level tiles not cured by dividing by the gcd are {0, 1, 8, 9} and {0, 3, 8, 11} at base 12, both mirror-symmetric. Witness: lab/py/level-tiles census.
• 2026-10-03 [Proved] If F satisfies T1 and T2 and lcm(S_F) divides b, then every level satisfies T1 and T2 and tiles the integers. Witness: notes/tilings.md "T2 at every level".
• 2026-10-03 [Proved] Under the T1 condition, T2 holds at every level if and only if it holds at levels 1 to 1 + G(2 omega - 1), with G the largest ceil(C_p/v_p(b)), C_p the largest exponent of p in an element of Z(F), and omega the number of primes with Z_p(F) nonempty. Witness: notes/tilings.md "T2 at every level".
• 2026-10-03 [Proved] {0, 1, 2, 6, 7, 8} at base 18 tiles the integers and satisfies T1 at every level, while its level 2 fails T2 at Phi_108 and does not tile. Witness: notes/tilings.md "T2 at every level", lab/py/level-tiles witness.
• 2026-10-03 [Verified] Among digit sets whose size has two prime factors and is below the base, at bases up to 30 with at most 200000 sets a cell, those that tile, satisfy T1 at every level and fail T2 at a level up to 6 number 0 of size 6 at base 12, 16 of size 6 at base 18, 0 of size 10 at base 20, 4 of size 6 at base 24 and 258 of size 6 at base 30, all failing at level 2, and 18 is the least base with such a digit set. Witness: lab/py/level-tiles witness.
• 2026-10-03 [Proved] By Coven-Meyerowitz Theorems A, B1 and B2 applied level by level through the index lemma and the horizon, every level of F at base b tiles the integers if and only if the T1 condition holds, when #F is a prime power, and if and only if the T1 condition holds and T2 holds up to the horizon, when #F has two prime factors; with three or more prime factors the same condition is sufficient. Witness: notes/tilings.md "The classification".
• 2026-10-03 [Proved] If T1 and T2 hold at every level of F, every level is spectral; in particular every level is spectral when every level tiles and #F has at most two prime factors. Witness: notes/tilings.md "Spectral levels".
• 2026-10-03 [Verified] On the 6866 levels with #A_n <= 36 at bases 2 to 12 and levels 1 to 3, a level is spectral exactly when it tiles on all 6749 levels the search decides; the 117 non-tiling levels of digit sets with a root on the unit circle that is not a root of unity stay undecided. Witness: lab/py/level-tiles spectral.
• 2026-10-03 [Conjecture] A level of a dimension-one design is spectral if and only if it tiles the integers. Witness: notes/tilings.md "Spectral levels", lab/py/level-tiles spectral.
• 2026-10-03 [Proved] For consecutive digits {0..N-1} at base b, every level tiles the integers if and only if N divides b, the criterion Dai, He and Lau give for the spectrality of the limit measure. Witness: notes/tilings.md "The limit measure".
• 2026-10-03 [Verified] For product-form digit sets {0..N-1} + m{0..L-1} at base p <= 24, 2 <= N, L <= 12, N <= m <= p^2, T1 and T2 at every level agree with the spectral condition of Liu-Wang-Zheng Theorem 1.3, an unrefereed arXiv preprint, on all 576422 cases. Witness: lab/py/level-tiles measure.
• 2026-10-03 [Verified] The digit set {0, 1, 8, 9} at base 12 meets the spectral condition of Liu-Wang-Zheng Theorem 1.3, an unrefereed arXiv preprint, with N = L = 2, m = 8, d = 1, while no integer multiple of it tiles Z/12 by the exponent argument and the exact-cover search over v = 1..12. Witness: notes/tilings.md "The limit measure", lab/py/level-tiles measure, witness.
• 2026-10-03 [Conjecture] For an integer base b >= 2 and a finite digit set F of non-negative integers containing 0, the self-similar measure mu_{b,F} is spectral if and only if every level is a set that tiles the integers. Witness: notes/tilings.md "The limit measure", lab/py/level-tiles measure.