Complex dimensions

The core page gives every design one real number, the dimension log(fill)/log(n). Fractal-string theory promotes that number to the real part of an infinite family — the complex dimensions, the poles of a zeta function attached to the set's gaps. This page computes them for the 1D designs, watches the imaginary parts surface as an oscillation in the box count, and follows the theory to its structural consequence, which needs more qualifying than the research notes gave it.

Every claim carries a tag. Proved means a proof is given or restated here; Verified means recomputed from scratch for this page; Conjecture means neither. Everything was run twice, by two full passes sharing no code: lab/lattice/poles.py, boxcount.py, tube.py and the independent rebuild lab/lattice/zeta.py, oscillate.py, content.py, with logs and RUN.txt.

The string of a design

A 1D design at base n is a subset F of the n digits; it draws the set of x in [0,1] whose base-n digits all lie in F. That set is the attractor of k = |F| maps x -> (x + f)/n, every one with the same contraction ratio 1/n, and its dimension is d = log(k)/log(n). The core page's parity rule at base 3 fills digits {0,2} — the middle-thirds Cantor set, d = log(2)/log(3) = 0.630930.

The complement of the set is a multiset of gaps, and the gap multiset of a one-base design satisfies G = G_1 + k copies of G/n. So its geometric zeta function — the sum of g^s over all gaps — has the closed form

zeta(s) = D(s) / (1 - k*n^(-s)),    D(s) = sum of g^s over level-1 gaps,

and its poles sit where the complex Moran equation k*n^(-s) = 1 holds:

s = d + 2*pi*i*m/ln(n),    m in Z.

One vertical line of poles, equally spaced at omega = 2*pi/ln(n). Sets whose complex dimensions line up on such an arithmetic progression are called lattice; sets with incommensurable ratios, whose poles spread out, are nonlattice. Every one-base design is lattice for the trivial reason that all its ratios are equal.

Verified. The progression is derived in closed form above and then evaluated: for the Cantor design and three others (table below), all 81 predicted poles at m = -40..40 kill the denominator to 5e-14, and the numerator D(s) stays bounded away from zero at every one of them — minimum |D(s)| = 0.500000 for the Cantor design — so no zero is cancelled and every one is a genuine pole. The identity D(1) = 1 - k/n, the statement that the gaps of a measure-zero set fill the whole interval, holds for all four.

One caveat the notes did not carry: the pole set is the zero set of the denominator only where the numerator does not vanish, and a design with no gaps (k = n), with one filled digit (k = 1, a point), or with none has an empty gap multiset, an identically zero zeta function, and no complex dimensions at all.

The oscillation in the box count

The imaginary parts are not bookkeeping; they are visible. Let N(eps) be the number of eps-cells the set meets and detrend it: g(u) = ln N(exp(-u)) - d*u. A lattice set's g oscillates at angular frequency omega = 2*pi/ln(n) forever.

Verified. For all four objects the periodogram peak of g(u) lands within 1% of the predicted 2*pi/ln(n) — under every box counter, window and padding tried across the two passes; for the Cantor design the readings run 5.688 to 5.741 against the predicted 5.719202. The four-decimal peaks the research notes quote (5.7120, "0.13% off") sit inside one frequency bin of their own estimator's resolution and are properties of the estimator, not of the object; what is stable, and what ships, is the sub-1% agreement.

A second reading folds u modulo each candidate period and asks how much of the variance of g the folded profile explains. At 40 bins on a fixed window:

Verified. Each lattice object folds best at its own ln(base), and the two controls fold well at none. But the two lattice signatures are not parallel in strength, and the research notes' prose implied they were: the base-15 design's folding explains 66.7% of the variance at its period, the Cantor design's only 19.2% at its own. The Cantor figure is genuinely modest — the box count is a step function and grid alignment injects a large aperiodic component — and it moves between roughly 0.16 and 0.23 as the bin count and window vary, so it is also partly a property of the estimator. The ordering is stable; the percentages are not constants of the objects.

Composition multiplies the base

The research notes compose two rules by alternating bases across levels: base 3 with digits {0,2} at odd levels, base 5 with digits {0,4} at even ones.

Verified. The alternation produces exactly the one-base design at base 15 with digits {5*d1 + d2} = {0,4,10,14} — checked as integer arithmetic and then as geometry, eight alternating levels and four base-15 levels producing the identical 256 intervals as exact fractions. So the composite is a different lattice period, omega = 2*pi/ln(15) = 2.320188, not a departure from the lattice class — and the table above shows its signature is the sharpest of the family, 66.7% of variance at its own period.

Two corrections to the notes, both load-bearing. First, base 5 with digits {0,4} is not a mrly design: the core page's move one fills by parity, and the even digits of base 5 are {0,2,4}, three of them — nor is the base-15 composite a parity design. The parity-faithful versions — base 5 {0,2,4} and their composite base 15 {0,2,4,10,12,14} — were run alongside and behave identically, which is the real point: the argument turns on the base, not on which digits the rule picks.

Second, the notes' structural claim — one contraction ratio per level implies lattice — is false without a periodicity hypothesis. An aperiodic control that alternates bases 3 and 5 on a Thue-Morse schedule uses exactly one ratio per level and is not self-similar at all, so it is neither lattice nor nonlattice: its best folding is 0.115 against the composite's 0.667, and its periodogram peak matches no 2*pi/ln(base). What survives, and needs no computation, is the statement for mrly designs proper: move two is a Kronecker power of one tile, so the schedule is constant, every level subdivides by the same n, and periodic cross-base alternation multiplies into one product base. Neither move can express two ratios inside one level — move one only chooses which cells of a fixed n^D grid survive, and every cell of that grid is the same size. (Verified for the alternation; the one-tile argument is read off the definition.)

The genuine way out is a two-ratio system: maps of ratio 1/3 and 1/5 mixed within one level, outside the mrly family, with d = 0.518370 solving 3^(-d) + 5^(-d) = 1. Verified. Its 21 complex dimensions in the box Re in [-3,3], Im in [-40,40] — a complete list, by the argument principle: the winding number over the box is 21 and 21 roots were found — have real parts spread from -0.699926 to 0.518370 and fit no arithmetic progression, the worst offset being 0.43, 0.17 and 0.38 of a step for the three candidate spacings. Nonlattice is a real, different behaviour, and no mrly design or composition exhibits it.

Measurability, with its hypotheses

A set is Minkowski measurable when M(eps) = eps^(d-1) * V(eps) — V the inner tube, the length of the set's eps-neighbourhood inside the gaps — has a limit as eps -> 0. The lattice/nonlattice split decides this, and the research notes overstated what is known.

Proved. The Cantor design {0,2} at base 3 is not Minkowski measurable. Splitting the tube sum at the scale of eps gives the exact limit profile

M -> 2^(1-d) * (t^(d-1) + t^d),    t in [1/3, 1),

one fixed profile traversed each time eps is divided by 3, with minimum 2.494975716 at t = (1-d)/d = 0.584963 and maximum 2.583040469 at the ends — a swing of 3.53%, so the profile is not constant and the limit does not exist. The measured tube matches the closed form to 4.9e-9 at the minimum.

Verified. The other three lattice objects behave the same way: the swing of M(eps) over successive windows is flat from u = 15 out to u = 60 (eps = 8.8e-27), and each object satisfies M(eps) = M(eps/n) at its own base to 1e-9 or better and at neither other candidate. The two-ratio control does the opposite: its swing decays monotonically 3.79% to 0.42% and is still falling — converging, as the nonlattice side predicts.

The literature, read rather than recalled, is not symmetric. Nonlattice self-similar sets under the open set condition are Minkowski measurable in every dimension (Gatzouras 2000). Lattice sets are not — but as a theorem only on the line, for a nontrivial set of non-integer dimension (Falconer 1995, completed by Kombrink and Winter 2020; for self-similar strings, Lapidus and van Frankenhuijsen 2006 — the book itself could not be opened for this page, so that attribution rests on secondary citations, said plainly in RUN.txt). In dimension 2 and above the lattice direction is an open conjecture of Lapidus, proved under a pluriphase hypothesis and for particular families, open in general.

So the notes' closing claim — every mrly design is not Minkowski measurable — ships only with two qualifications, and both bite on real designs.

1. Nontrivial fill and non-integer dimension. The theorem's own hypotheses

exclude integer dimension, and the exclusion is not exotic: mrly_03, the core page's pin(y), fills 2 of 4 at base 2, has d = log(2)/log(2) = 1 exactly, draws a segment — and a segment is Minkowski measurable. The solid, single-point and empty designs have no gaps and no oscillation to have. At base 3 in 1D the only design of non-integer dimension is {0,2} itself. (Verified: the counterexample and the census of which designs the statement covers.)

1. Dimension one only. For 1D designs with 2 <= k < n and non-integer

d, non-measurability is a theorem, and for the Cantor design it is proved outright above. For the carpet and the sponge, lattice membership holds — but non-measurability is the conjectured consequence, a Conjecture here as in the literature, not a citable theorem.

Where the numbers live

lab/lattice/ holds the two independent passes — poles.py, boxcount.py, tube.py and zeta.py, oscillate.py, content.py — their logs, and a RUN.txt recording what reproduced, what is estimator-dependent, and where each method stopped. The dimension formula this page extends, and the designs it names, are the core page; the spectral side of the same fractals is the complexity page.