multilayers.md

A design at base b with digit set D = {d_0 < ... < d_(m-1)} is the set of integers whose k base-b digits all lie in D, the one-dimensional missing-digit set and at b = 3, D = {0,2} the Cantor set at level k. Lay glass on it: cell p of a row of b^k cells is layer A (index nA) when p is in the design and host B (index nB = 1.45) otherwise, every cell of the same optical thickness, so light at normal incidence picks up the same phase delta crossing any cell. The stack sits in B on both sides; r_k and t_k are its reflection and transmission amplitudes, T_k = abs(t_k)^2, and the stack is open at delta when abs(r_k/t_k) < 1, that is T_k > 1/2. The passband fraction f_k is the share of delta in (0, pi) where the level-k stack is open.

Two polynomials carry the design. The digit polynomial P_D(z) = sum_(d in D) z^d, and the block-spacer lift Q_D(x, y) = sum_i x^i y^(d_i - i), where x stands for one level-k block and y for one empty spacer of the same length; Q_D(z, z) = P_D(z). Their Mahler measures are m(P) = integral ln abs(P) over the unit circle and the torus. The generator for every number below is lab/py/design-multilayer; its verbs are named where they print.

The level recursion

The level-(k+1) characteristic matrix is M_(k+1)(delta) = W_D(M_k(delta), M_B(b^k delta)), the ordered product over d = 0 .. b-1 of M_k when d is in D and of the plain host slab M_B(b^k delta) otherwise. Proved. The level-(k+1) word is b blocks of length b^k whose leading digit is d: the block is the level-k word when d is in D and b^k host cells otherwise, the transfer matrix of a concatenation is the product of the parts, and b^k host cells are one host slab of phase b^k delta. It is the Kronecker product read as a substitution, and it costs b - 1 matrix multiplications a level, so 7^12 layers cost 72. Exact self-similar algorithms of this kind go back to Jaggard and Sun 1990 for Cantor-bar multilayers.

The recursion agrees with the layer-by-layer product to 1e-12 relative. Verified (check): the brute-force product runs in 40-digit arithmetic over 3^6, 4^5, 5^4, 7^4 and 9^3 cells at nA = 1.6, 2.3, 3.5, eight frequencies each, and the largest relative gap is 8.4e-13.

The Born drift

At zero contrast the stack's reflection has the design's generating function as its derivative: abs(d(r_k/t_k)/d nA) at nA = nB equals abs(d(r_0/t_0)/d nA) times abs(prod_(j<k) P_D(e^(2i b^j delta))), so the log of its modulus, averaged over delta uniform in (0, pi), grows by exactly m(P_D) a level. Proved. At nA = nB the stack is homogeneous and r = 0; the first derivative of the product in nA is a sum over A cells of one perturbed factor, each A cell at optical position p delta contributing the single-cell derivative times e^(2i p delta) up to one common phase, and the positions are the digit sums, whose generating function is prod_(j<k) P_D(z^(b^j)). Each factor's mean log modulus is m(P_D) because b^j delta is uniform whenever delta is. check confirms the identity at nA = nB(1 + 1e-8) to 5e-7 in abs(gap)/max(1, abs(prod)) on three designs through level 4.

This is a statement about the zero-contrast derivative and nothing more; the true stack follows it only while nA - nB times the product stays small. At nA = 1.455 the mean per-level drift of ln abs(r_k/t_k) over six to eight levels is within 0.001 of m(P_D). Verified (born): 0.3832 against m(P_D) = 0.38225 for {0,1,3} base 4, 0.4425 against 0.44214 for {0,1,2,4} base 5, 0.2818 against 0.28120 for {0,1,5} base 7, and within 0.004 of 0 for the three designs with cyclotomic P_D, standard errors below 0.002. The agreement is to leading order, not to noise: 0.3832 sits about three standard errors off. The Born picture is the diffraction picture: for binary constant-length substitutions the Mahler measure of a Borwein polynomial is a maximal Lyapunov exponent of the Fourier matrix cocycle (Baake, Coons and Manibo 2020).

The block-spacer lift

At fixed contrast the Born picture breaks within a few levels, and the reason is phase. A level-k block that is nearly transparent acts on light as a phase u = t_k/abs(t_k), and at zero contrast u equals the spacer's phase v = e^(i b^k delta); at finite contrast the two part. To first order in r_k, the next level's reflection is the sum of the m block echoes, the i-th reached through i blocks and d_i - i spacers, so r_(k+1) = r_k Q_D(u^2, v^2) up to a phase and a relative error of order abs(r_k). The digit polynomial is the diagonal u = v of that law.

The step law ln abs(r_(k+1)/t_(k+1)) - ln abs(r_k/t_k) = ln abs(Q_D(u^2, v^2)) + O(abs(r_k)). Verified (deep): at every level where abs(r_k/t_k) < 1e-3 on at least 50 frequencies, the median gap between the two sides is below 3e-6, on the nine deep designs (the seven of the table, {0,1,3} base 4 and {0,1,2,4} base 5) at nA = 1.6, 2.3, 3.5. That covers 25 of the 27 design-contrast cells; {0,1,3} base 4 and {0,1,2,4} base 5 at nA = 3.5 never reach 50 such frequencies. The first-order expansion above is the reason; it is not written out here as a proof.

For a digit set in arithmetic progression, Q_D depends on one monomial and the level step is exact at any contrast: one period of the progression is a block and its spacer, and the Abeles formula for a periodic stack gives abs(r_(k+1)/t_(k+1)) = abs(U_(m-1)(chi_k)) abs(r_k/t_k) with chi_k the half-trace of one period and U the Chebyshev polynomial of the second kind. It is the textbook periodic-stack formula; Chebyshev polynomials of the second kind also carry the exact transmission of generalised Cantor-like potentials in Ogawana and Sakaguchi 2018.

Cyclotomic is not enough

The Born drift vanishes exactly when P_D is a product of cyclotomic polynomials (Kronecker's theorem, the one-variable case of Boyd's), and the natural guess is that such a stack is critical: no drift, passband lost only slowly. The step law says the guess reads the wrong polynomial.

"P_D cyclotomic implies a critical stack." Refuted (refute) by {0,2,3,4,6} base 7, {0,1,4,7,8} base 9 and {0,2,3,4,5,7} base 8. For each, P_D factors into cyclotomic polynomials over the integers, while Q_D is not a monomial times cyclotomic polynomials in monomials, the exact test run by factoring in two variables, so m(Q_D) > 0 by Boyd's theorem (Boyd 1981): m(Q_D) = 0.2513, 0.2513, 0.3181. At nA = 2.3 their f_k sqrt(k) falls without pause from level 2 to level 12, 0.878 to 0.548, 0.883 to 0.538 and 0.888 to 0.406, while the controls {0,2,4,6} base 7 and {0,1,3,4} base 7, whose Q_D is Boyd-cyclotomic, hold at 1.097 and 0.901. Their deep drift, below, is at least 0.26 a level at all three contrasts.

The deep drift

If the pair (u, v) were spread evenly over the torus on the passband, the step law would make the drift m(Q_D), the two-variable Mahler measure. It is close, and at weak contrast it is not equal.

The table is deep: the deep drift is the mean step over frequencies with abs(r_k/t_k) < 0.02, averaged over the last four measured levels, and se is the mean of their four standard errors, which bounds the standard error of that average; each cell lists nA = 1.6, 2.3, 3.5, on 2e5 frequencies.

"At nA = 1.6 the drift of ln abs(r/t) deep in the passband tends to m(Q_D)." Refuted (deep): the deep drift is 0.350 against m(Q_D) = 0.3181 for {0,2,3,4,5,7} base 8 and 0.272 against 0.2513 for {0,2,3,4,6} base 7, 6.3 and 4.3 standard errors above, each flat over its last four levels. At nA = 2.3 and 3.5 the gaps of these three designs are 2.6 to 4.6 standard errors on series that still move ({0,2,3,4,6} base 7 falls from 0.327 to 0.285 at nA = 3.5), so there the limit is open.

The torus mean is ln abs(Q_D(u^2, v^2)) averaged over every frequency at the last level, dark or open; it starts at m(P_D), where u = v, and moves toward m(Q_D) as the phases part. At nA = 1.6 it ends on m(Q_D) to two digits in all four positive rows, consistent with (u, v) covering the torus evenly over all frequencies; at nA = 2.3 and 3.5 it holds flat below m(Q_D) on the three refutation designs. The deep drift is the same kind of average taken only where the stack is still transparent, and there the mean lift ln abs(Q_D(u^2, v^2)) tracks the mean step to 0.003 over the last four levels (deep), so the gap from m(Q_D) is the law of (u, v) on the passband, not the step. What survives is the zero set. The deep drift is at most 0.01 in absolute value, standard error at most 0.01, at all three contrasts on the three designs whose Q_D is Boyd-cyclotomic, and at least 0.26, standard error at most 0.02, on the four whose Q_D is not. Verified (deep).

For {0,1,3} base 4, Q_D = 1 + x + x^2 y, and the unimodular change y -> x^(-2) y makes it 1 + x + y, so m(Q_D) = 3 sqrt(3) L(chi_(-3), 2)/(4 pi) = 0.3230659472, Smyth's constant (Smyth 1981). m(Q_(0,1,3)) computed on the torus equals Smyth's constant to 1e-8. Verified (smyth, against PARI's lfun(-3, 2)). "At nA = 1.6 the {0,1,3} base 4 stack drifts at Smyth's constant." Refuted (deep): the deep drift over levels 8 to 11 is 0.51 to 0.60, standard error at most 0.026, while the torus mean over all frequencies ends at 0.323. At nA = 2.3 and 3.5 the deep set falls below 300 frequencies by level 3, too soon for a drift.

Two regimes

Conjecture: when m(Q_D) = 0 the passband fraction falls as f_k ~ C/sqrt(k) = C/sqrt(log_b L) in the number of cells L = b^k; when m(Q_D) > 0 it falls geometrically in k, a power of L. By Boyd's theorem m(Q_D) = 0 exactly when Q_D is a monomial times cyclotomic polynomials in monomials, which covers the progressions and sum sets such as {0,1,3,4}, where Q_D = (1 + x)(1 + x^2 y). The heuristic is the step law with a driftless walk: ln abs(r/t) performs a walk of mean 0 and the stack stays open while the walk stays below 0, which a centred walk does for k steps with probability of order 1/sqrt(k).

Every design at bases 3 to 6 sorts by the Boyd class. Verified (census): the 39 designs with 0 in D, 2 <= abs(D) < b, up to shift and mirror, at nA = 2.3 and 1e5 frequencies. The flatness f_12 sqrt(12)/(f_6 sqrt(6)) is 0.921 to 1.036 on the 26 Boyd-cyclotomic designs and 0.057 to 0.315 on the other 13; the verdict is critical above 0.9, below the predicted value 1, and it matches the class on all 39. At these bases no design has P_D cyclotomic with Q_D not, so the census cannot tell m(P_D) from m(Q_D); the three designs of the refutation do. At level 12, f_k sqrt(k) is 1.097 for {0,2,4,6} base 7, still falling slowly from 1.108 at level 6, and 0.901 for {0,1,3,4} base 7 (refute); all of this is at the one contrast nA = 2.3.

Open

• The joint law of the block phase u and the spacer phase v: the step law is exact to first order, and every gap above sits in how (u, v) spreads on the torus over the passband; a proof of either regime needs that law.
• The deep drift at nA = 2.3 and 3.5 on the three refutation designs: it still moves at the last measured level, so whether it tends to m(Q_D) there is not settled.
• The critical walk is a dependent walk driven by delta -> b delta; a Sparre Andersen bound for it would prove 1/sqrt(k) for the progressions, where the step is exact.
• The question the page hands to optics: which design with m(Q_D) > 0 darkens slowest. The smallest m(Q_D) > 0 in the census is Smyth's 0.3231 and the refutation designs reach 0.2513; whether designs can drive it toward 0, or toward Lehmer's constant, is Lehmer's problem restricted to staircase 0/1 polynomials, read as the slowest-localising fractal mirror; the one-variable problem already has a spectral reading in Baake, Coons and Manibo 2020.
• Related pages: crop and weights for the digit transform whose Riesz product is the Born drift, and the Rajchman property and the random walk for the two regimes.