Measure-theoretic preliminaries

We use the bit form \[X_\lambda=\sum_{i=0}^\infty u_i\lambda^i, \qquad \mathbb P(u_i=0)=\mathbb P(u_i=1)=\tfrac12,\] with law \(\mu_\lambda\), and write \(L=(1-\lambda)^{-1}\). The affine identity \(Y_\lambda=2X_\lambda-L\) shows that \(\mu_\lambda\) and \(\nu_\lambda\) have the same measure type. For a word \(u\in\{0,1\}^n\), put \[ s_u(\lambda)=\sum_{i=0}^{n-1}u_i\lambda^i. \tag{5}\] Conditioned on this prefix, \(X_\lambda\) lies in \(s_u(\lambda)+\lambda^n[0,L]\).

We shall also use the following elementary formulation of concentration. It isolates the measure-theoretic input needed to establish membership in (4).

Algebraic clusters imply singularity

Suppose \(\lambda\in(1/2,1)\) satisfies the condition in (4), and fix a degree multiplier \(d\) witnessing it. For each \(j\), there are arbitrarily large \(n\) and \(r\in R_*(n,d,j)\) such that \[r-4^{-n}\leq\lambda<r.\] If \(w\in W_n(r;d)\), choose a polynomial \(p_q\) giving this projection. Its root \(q\) satisfies \(|q-\lambda|\leq2\cdot4^{-n}\). On any fixed compact subinterval of \((0,1)\) containing \(\lambda\) in its interior, \[|p_q'(t)|\leq\sum_{k\geq1}k t^{k-1}=\frac1{(1-t)^2}.\] The coefficients of index at least \(n\) contribute at most \(L\lambda^n\) at \(\lambda\). Thus there is a constant \(B=B(\lambda)\), independent of \(n,d,j,w\), such that for all sufficiently large \(n\), \[ \left|\sum_{i=0}^{n-1}w_i\lambda^i\right|\leq B\lambda^n. \tag{6}\] Here we used \(4^{-n}=o(\lambda^n)\).

If \(w\in\mathcal O_u\cap\{-1,0,1\}^n\), then \(v=u-w\) belongs to \(\{0,1\}^n\). Different \(w\) give different \(v\). By (3), at least \(2^n/d\) words \(u\) consequently have at least \(j(2r)^n\geq j(2\lambda)^n\) neighbors \(v\) satisfying \[|s_u(\lambda)-s_v(\lambda)|\leq B\lambda^n.\]

Partition the line into half-open bins of length \(\lambda^n\), and let \(\mathcal I\) be the bins containing at least one of these words \(u\), where a word occupies the bin containing its prefix sum. For each \(J\in\mathcal I\), choose one such word. Its neighbors lie in the \(B\lambda^n\)-enlargement of \(J\). Each prefix sum can belong to the enlargements of at most \(N_B=2\lceil B\rceil+3\) bins. Counting word–bin incidences yields \[\#\mathcal I\,j(2\lambda)^n\leq N_B2^n, \qquad \#\mathcal I\leq\frac{N_B\lambda^{-n}}j.\] Enlarge each bin \(J=[a,a+\lambda^n)\) to \([a,a+(1+L)\lambda^n]\), and denote their union by \(E_j\). The tail bound following (5) gives \[ \operatorname{Leb}(E_j)\leq\frac{(1+L)N_B}{j}, \qquad \mu_\lambda(E_j)\geq\frac1d. \tag{7}\] Absolute continuity of a finite measure makes its mass uniformly small on sets of sufficiently small Lebesgue measure. Therefore (7), as \(j\to\infty\), rules out absolute continuity. Lemma 2 proves singularity.

Overlap of the first-bit measures

Define subprobability measures \[\mu_i(A)=\mathbb P(X\in A,\ u_0=i),\qquad i=0,1.\] Their sum is \(\mu\). The following fact uses only \(\lambda>1/2\), not singularity.

Completion to irreducible polynomials

For every pair in Proposition 5, the difference \(w=u-v\) has \(w_0=-1\) and \(|\sum_{i<n}w_i\lambda^i|\leq(L-a)\lambda^n\). We next extend any such coefficient vector to a polynomial whose root lies in \((\lambda,\lambda+4^{-n}/2]\). The first lemma ensures that a bounded-coefficient polynomial cannot remain too flat throughout this short interval.

To impose irreducibility without disturbing the prescribed coefficients, we use the prime-polynomial theorem in arithmetic progressions. We include the deduction needed here from the function-field Riemann hypothesis; its character-sum formulation is recalled in (Gorodetsky and Kovaleva 2024, Theorem 3.2 and Lemma 3.3).

Proof of the converse and the quantifiers

Suppose \(\mu_\lambda\) is singular. Proposition 5 gives constants \(a,c>0\). Choose one integer \(d\) large enough for Proposition 8 and also satisfying \(1/d<c\). This choice depends on \(\lambda\) alone.

Fix any integer \(j\geq1\) and choose \(Q>2j\). For all sufficiently large \(n\), each of at least \(c2^n\) words \(u\) has at least \(Q(2\lambda)^n\) neighbors \(v\) supplied by Proposition 5. For each difference \(w=u-v\), Proposition 8 supplies a root \(r_w\in R(n,d)\) with signed minimal polynomial projecting to \(w\), and \[\lambda<r_w\leq\lambda+\tfrac12 4^{-n}.\] Choose any one of these roots as a center \(r\). All of them lie within \(4^{-n}\) of \(r\), so all their projected vectors belong to \(W_n(r;d)\). For fixed \(u\), its different neighbors give distinct vectors \(u-v\), each in \(\mathcal O_u\). Furthermore, \[1\leq\left(\frac r\lambda\right)^n \leq\left(1+\frac{4^{-n}}{2\lambda}\right)^n\longrightarrow1\] uniformly in the choice of center. Hence, for all sufficiently large \(n\), each of these \(c2^n\) orthants contains at least \(j(2r)^n\) distinct vectors of \(W_n(r;d)\). Since \(c>1/d\), we have \(r\in R_*(n,d,j)\) and \(\lambda\in[r-4^{-n},r)\).

We have proved that a single \(d\) works for every \(j\), at every sufficiently large \(n\). This proves membership in the expression (4) with \(\liminf\). Membership in that expression implies membership with \(\limsup\), which implies singularity by Section 2.2. Lemma 2 handles \(\lambda\leq1/2\) and supplies the absolute continuity of every remaining parameter. This completes the proof of Theorem 1.

Multipliers with small unit-circle conjugates

Fix \(\beta\) as in Theorem 11, and write its conjugates as \(\beta^{-1},z,\bar z\), where \(z=e^{i\theta}\). The number \(\beta\) is a unit: the product of its four conjugates is \(1\). Moreover, \(\theta/\pi\) is irrational, since otherwise \(z\), and hence its conjugate \(\beta\), would be a root of unity.

For positive integers \(k_1,k_2,\ldots\), to be chosen, define \[ \begin{aligned} A_m&=\prod_{h=1}^m(\beta^{k_h}-\beta^{-k_h}), &b_m&=\sum_{h=1}^m k_h,\\ c_m&=\prod_{h=1}^m(z^{k_h}-z^{-k_h}), &\delta_m&=|c_m|,\qquad L_m=\log(1/\delta_m). \end{aligned} \tag{16}\] Here and throughout this section logarithms are natural. Irrationality ensures \(\delta_m>0\), and \(0<A_m\le\beta^{b_m}\). For \(j\in\mathbb Z\), put \(u_{m,j}=A_m\beta^j\). All these numbers are algebraic integers, because \(\beta\) is a unit. Their field traces are the integers \[ T_{m,j}=u_{m,j}+(-1)^m u_{m,-j} +2\operatorname{Re}(c_mz^j)\in\mathbb Z. \tag{17}\] Indeed, the embedding taking \(\beta\) to \(\beta^{-1}\) takes \(A_m\) to \((-1)^mA_m\). Also \(u_{m,j}\notin\mathbb Z+\tfrac12\), since a rational algebraic integer is an integer. Thus none of the cosine factors used below vanishes.

The construction below balances the decrease of \(\delta_m\) against the cosine losses generated by \(A_m\). Eventually \(L_m\) grows by a factor greater than two while \(k_{m+1}\delta_m^2\) tends to zero; these are the margins used in the loss estimate below. It also keeps \(b_m\) small compared with \(\delta_m^{-1}\), leaving room for many separated blocks of signs. A uniform lower bound on the cosines will permit a quadratic estimate for perturbations of their logarithms.

The loss in the cosine products

Fix the sequence furnished by Lemma 12. Constants denoted by \(C\) below may change from line to line, but are independent of \(m\) and of the phase index. Define \[f(u)=-\log|\cos(\pi u)| \quad\text{for }u\notin\mathbb Z+\tfrac12, \qquad S_m=\sum_{j\le0}f(u_{m,j}).\] Each \(S_m\) is finite: its terms have a geometrically decaying tail because \(u_{m,j}\to0\) as \(j\to-\infty\) and \(f(u)=O(u^2)\) near zero. The next estimate controls the entire negative-power contribution even though \(A_m\) becomes large.

Independent blocks of Fourier phases

We now turn the product estimates into sets of almost full \(\nu_{1/\beta}\)-measure and vanishing Lebesgue measure. The comparison of phase averages under two measures is analogous to the separation argument for Riesz products in (Peyrière 1975, sec. 1.1, Theorem 1.2). Here the needed estimates follow from the independent signs, as we now show. Work on the probability space of independent uniform signs \((\epsilon_\ell)_{\ell\ge0}\), and write \[Y=\sum_{\ell\ge0}\epsilon_\ell\beta^{-\ell},\qquad W_{m,n}=\exp(i\pi A_m\beta^nY)\quad(n\ge0).\] The law of \(Y\) is \(\nu_{1/\beta}\), supported on \(I=[-(1-\beta^{-1})^{-1},(1-\beta^{-1})^{-1}]\). Independence, followed by passage to the limit in the convergent sign series, gives \[ \mathbb EW_{m,n}=\prod_{j=-\infty}^n\cos(\pi u_{m,j}). \tag{29}\] This is a nonzero real number; the negative-index tail has a convergent sum of logarithmic losses.

Set \[J_m=\lfloor e^{1.5L_m}\rfloor,\qquad H_m=\lceil b_m+DL_m\rceil,\] \[\mathcal N_m=\{H_m+s(2H_m+1):s\in\mathbb Z_{\ge0}\}\cap[0,J_m],\] and let \(M_m=|\mathcal N_m|\). All subsequent estimates concern sufficiently large \(m\). Since \(H_m\le K_m\le e^{.75L_m}\), \[ e^{.7L_m}\le M_m\le J_m+1. \tag{30}\] Lemma 13 and \(J_m\delta_m^2\le e^{-.5L_m}\) show that \[ |\mathbb EW_{m,n}|\ge e^{-L_m/20} \quad(0\le n\le J_m). \tag{31}\]

For \(n\in\mathcal N_m\), retain only the signs whose indices lie in \([n-H_m,n+H_m]\), and define \[ \widetilde W_{m,n} =(-1)^{\sum_{j=H_m+1}^nT_{m,j}} \exp\left(i\pi\sum_{\ell=n-H_m}^{n+H_m} u_{m,n-\ell}\epsilon_\ell\right). \tag{32}\] The integer sum in the prefactor is empty when \(n=H_m\). To explain this approximation, expand the exponent of \(W_{m,n}\) as \(\pi\sum_{j\le n}u_{m,j}\epsilon_{n-j}\). For \(j>H_m\), replace \(u_{m,j}\) by its integer trace \(T_{m,j}\); the resulting factor is deterministic because \(e^{i\pi T_{m,j}\epsilon_{n-j}}=(-1)^{T_{m,j}}\). For \(j<-H_m\), drop the term. The remaining indices are exactly those in (32).

The difference of the two phase angles, after these replacements, is a sum of independent mean-zero signs. The inequality \(|e^{ix}-e^{iy}|\le|x-y|\) therefore gives \[\begin{align*} \|W_{m,n}-\widetilde W_{m,n}\|_2^2 &\le\pi^2\left( \sum_{j=H_m+1}^n|u_{m,j}-T_{m,j}|^2 +\sum_{j<-H_m}|u_{m,j}|^2\right)\\ &\le C\left(\beta^{2(b_m-H_m)}+n\delta_m^2\right) \le Ce^{-L_m/2}. \tag{33}\end{align*}\] Here (17) bounds the first summand by a geometric tail and a contribution \(C n\delta_m^2\); the other sum is itself a geometric tail. Finally, \(D\log\beta=10\) and \(n\le J_m\) give the last inequality. All norms in (33) refer to the sign probability space. The windows belonging to consecutive members of \(\mathcal N_m\) are disjoint, so the variables \(\widetilde W_{m,n}\) are independent.

Two explicit parameters

Word vectors and characters

For a word \(a=(a_0,\ldots,a_{N-1})\in\{0,1\}^N\), define \[ y_r(a)=\sum_{j=0}^{N-1}a_jr^j\quad(r\in D\cup I), \qquad x_r(a)=r^{-N}y_r(a)\quad(r\in D). \tag{40}\] Vectors indexed by roots are always constrained by \(y_{\overline r}=\overline{y_r}\), and a conjugate pair counts as one complex coordinate, or two real coordinates. In particular, the space of all \(y\)-vectors has real dimension \(31\).

To distinguish the word vectors from typical points of these boxes, define the real sequences \[ C_0(m)= \begin{cases} \displaystyle\sum_{r\in I}\frac{r^{-1-m}}{p'(r)},&m<0,\\[4pt] \displaystyle-\sum_{r\in D}\frac{r^{-1-m}}{p'(r)},&m\ge0, \end{cases} \qquad C_1(m)=C_0(m+3)-C_0(m)+C_0(m-1). \tag{41}\] Partial fractions give \[\frac1{p(t)}=\sum_{m\in\mathbb Z}C_0(m)t^m, \qquad \max_{r\in I}|r|<|t|<\min_{r\in D}|r|.\] Thus \(C_0\) is the bilateral Laurent coefficient sequence of \(1/p\), and both \(C_0,C_1\) decay exponentially at both ends. The same inversion by an exponentially decaying two-sided sequence is used by Lind and Schmidt (Lind and Schmidt 1999, Lemma 4.5 and Example 4.7) to construct homoclinic points for algebraic actions. Here the explicit root formula will give characters adapted to the word vectors.

For arbitrary expanding coordinates \(x_D\), with \(x_\beta\in\mathbb R\) and \(x_{\overline\alpha}=\overline{x_\alpha}\), put \[ \begin{split} Z_i^{(0,N)}(x) &=\exp\!\left(-2\pi\mathrm i\sum_{r\in D} \frac{x_rr^i}{p'(r)}\right) \exp\!\left(-\pi\mathrm i\sum_{k=0}^{N-1}C_0(k-i)\right),\\ Z_i^{(1,N)}(x) &=Z_{i-3}^{(0,N)}(x)\overline{Z_i^{(0,N)}(x)}Z_{i+1}^{(0,N)}(x), \qquad i\in\mathbb Z. \end{split} \tag{42}\] These functions have modulus one.

The corresponding stationary model uses independent fair bits \((b_k)_{k\in\mathbb Z}\) and the unit random variables \[ \mathcal Z_i^{(A)}= \exp\!\left(\pi\mathrm i\sum_{k\in\mathbb Z}C_A(k-i)(2b_k-1)\right). \tag{44}\] The series converges absolutely. Independence gives the real moments \[ \begin{split} P_A&=\mathbb E\mathcal Z_i^{(A)} =\prod_{m\in\mathbb Z}\cos(\pi C_A(m)),\\ P_{AB}^{\pm}(k) &=\mathbb E\bigl[\mathcal Z_i^{(A)} (\mathcal Z_{i-k}^{(B)})^{\pm1}\bigr]\\ &=\prod_{m\in\mathbb Z}\cos\bigl(\pi(C_A(m)\pm C_B(m+k))\bigr), \qquad k\ge0. \end{split} \tag{45}\]

A predictor with a positive mean gain

The weakly expanding pair produces long decaying tails in \(C_0\). Some shifted tails nearly cancel in the signed products \(P_{00}^{\pm}(k)\). A linear combination of the corresponding phases will use these correlations. For rational weights \(w_{ks}\), define \[ \begin{split} G_i^{(N)}(x) &=-0.001+0.0055 Z_{i-33}^{(1,N)}(x) +\sum_{k=30}^{2000}\sum_{s=0}^1 w_{ks}\bigl(Z_{i-k}^{(0,N)}(x)\bigr)^{1-2s},\\ \mathcal G_i &=-0.001+0.0055\mathcal Z_{i-33}^{(1)} +\sum_{k=30}^{2000}\sum_{s=0}^1 w_{ks}(\mathcal Z_{i-k}^{(0)})^{1-2s},\\ \mathcal L_i&=\operatorname{Re}(\mathcal Z_i^{(0)}\mathcal G_i). \end{split} \tag{46}\]

We fix weights supplied by Proposition 19. Their exact rational construction is given in Section 7.2, and the moment bounds are proved in Section 7.4.

The next two subsections construct a positive product whose logarithm has a positive mean under this stationary model and whose integral is bounded in the expanding coordinates. These two estimates will remove an exponential proportion of the lattice points allowed by Lemma 17.

Damping the possible frequency cancellations

Choose a fixed integer \(i_0\ge1\) so that \[ \frac{0.1\beta^{i_0}}{|p'(\beta)|}>1. \tag{49}\] For a word length \(N\), use every complete block \[B_q=\{i_0+22q,\ldots,i_0+22q+20\}\subseteq\{0,\ldots,N-1\}, \qquad q\ge0,\] and write \(J_N\) for their union. Thus each block has \(21\) indices and consecutive blocks are separated by one unused index.

The near-relation (37) dictates which signed digit patterns need to be retained when integrating a block. For a pattern \((d_i)_{i\in B_q}\in\{-1,0,1\}^{B_q}\), let \(b=\max B_q\) and set \[v_i=\beta^{-i}\sum_{j=i}^{b}d_j\beta^j,\qquad v_{b+1}=0.\] Reading from the largest index downwards gives \(v_i=\beta v_{i+1}+d_i\). At the root \(\beta\), (37) becomes \[ \beta^3-\beta^2-\beta-1=\beta^{-30}. \tag{50}\] In particular, the four digits \([\sigma,-\sigma,-\sigma,-\sigma]\) leave the small normalized remainder \(\sigma\beta^{-30}\). We encode these approximate returns by the transitions in Figure 1. The initial state is \(0\), and the permitted terminal states are \(0,C_+,C_-\). For each sign \(\sigma\in\{-1,1\}\), the numerical values assigned to the states are \[ 0,\qquad A_\sigma:\ \sigma,\qquad B_\sigma:\ \sigma(\beta-1),\qquad C_\sigma:\ \sigma(\beta^2-\beta-1). \tag{51}\] A pattern is called admissible if every transition is allowed and its terminal state is permitted.

Let \(o(d)=\#\{i:d_i\ne0\}\). At a position \(i\) of a block, define \[ \eta_i=\sum_{\substack{d\text{ admissible}\\d_i\ne0}} \frac{x_*^{o(d)-2}}{o(d)}. \tag{52}\] The same coefficients are repeated in every block. An admissible nonzero pattern has at least three nonzero digits, so all summands are well defined.

For the moment fix \(N\), and abbreviate \(Z_i=Z_i^{(0,N)}\), \(G_i=G_i^{(N)}\), and \(x_i=|G_i|\). Define \[ h_i=1-\eta_i x_i^2,\qquad F_i=h_i\bigl(1+(1-x_i^2)(Z_iG_i+\overline{Z_iG_i})\bigr), \qquad i\in J_N. \tag{55}\] These factors are bounded above and bounded away from zero uniformly in \(N,i,x_D\). Indeed, \(h_i\ge1-S_*/3\), and the other factor is at least \(1-2x_*>0\).

A typical exponential gain

We now evaluate the same factors in the stationary model, replacing \(Z_i,G_i\) in (55) by \(\mathcal Z_i^{(0)},\mathcal G_i\); denote the resulting factors by \(\mathcal F_i\). The damping coefficients remain periodic with period \(22\), and are used at the same \(21\) positions per period.

Counting typical vectors and proving singularity

The integral bound applies to continuous coordinates, whereas Proposition 23 concerns lattice points. The remaining step is to thicken each such point by a fixed radius. Normalization in the expanding directions makes the resulting change in the whole logarithmic product bounded independently of the word length.

Locating all the roots

Recall the polynomial \[p(t)=\sum_{j=0}^{9}(-t^{3j}+t^{3j+1})-2t^{30}+t^{31}.\] Put \(E=10^{40}\). The entries in Table 1 are the real and imaginary numerators of centers \(z=(a+\mathrm i b)/E\). Include the conjugate of each nonreal center. Thus the table specifies \(31\) disks, all of radius \(\delta=10^{-35}\). Denote its first and sixth centers by \(z_\beta\) and \(z_\alpha\), respectively.

For a complex number write \(\|z\|_1=|\Re z|+|\Im z|\) and \(\|z\|_\infty=\max(|\Re z|,|\Im z|)\). Exact Horner evaluation gives, at every displayed center, \[ |z|<2,\qquad \|p'(z)\|_\infty>1,\qquad 2\cdot10^{35}\|p(z)\|_1<\|p'(z)\|_\infty. \tag{65}\] Here is an integer prescription for these comparisons. For \(q(t)=\sum_{j=0}^d q_jt^j\) and the Gaussian integer \(Z=Ez\), initialize \((V,h)=(q_d,1)\) and, for \(j=d-1,d-2,\ldots,0\), replace \[h\leftarrow Eh,\qquad V\leftarrow ZV+hq_j.\] The final value is \(q(z)=V/h\). Apply this to \(p\) and \(p'\), and clear the positive denominators in (65). Comparisons of squared distances give \(|z-z'|>.01\) for distinct centers, including conjugates. Comparisons of coordinates and squared moduli also give \[ \begin{gathered} 1.83928<z_\beta<1.83929,\qquad 1.0002090<|z_\alpha|^2<1.0002093,\\ |z|<.9939\quad\hbox{at all other displayed centers}. \end{gathered} \tag{66}\] For \(|h|=\delta\), the terms of degree at least two in \(p(z+h)-p(z)-p'(z)h\) have total modulus at most \[\delta^2\sum_{j=0}^{31}|p_j|3^j<10^{17}\delta^2.\] Indeed, \(|z|<2\) and \(\delta<1\), so the binomial expansion gives this bound term by term. By (65), the constant term is less than \(|p'(z)|\delta/2\), whereas the displayed remainder is less than \(|p'(z)|\delta/2\). Rouché’s theorem therefore puts exactly one root, counted with multiplicity, in each disk. The disks are disjoint and their number is the degree, so these are all the roots and all are simple. The root in the real-centered disk is real by conjugation. Every other disk misses the real axis, so the remaining roots form fifteen nonreal conjugate pairs. The radius \(10^{-35}\) and (66) imply every bound in Proposition 16, including \(|1/\alpha|<.9999\).

Rounded coefficient arrays and finite products

The numerical inputs are the sequences \(C_0,C_1\) defined in the main text in (41). We approximate their single and paired moments \[ P_A=\prod_{m\in\mathbb Z}\cos(\pi C_A(m)),\qquad P_{AB}^{\sigma}(k)= \prod_{m\in\mathbb Z}\cos\bigl(\pi(C_A(m)+\sigma C_B(m+k))\bigr), \tag{67}\] where \(A,B\in\{0,1\}\), \(\sigma\in\{1,-1\}\), and \(k\ge0\). Both products converge by exponential decay of the coefficients. The sign notation \(+\) and \(-\) means \(\sigma=1\) and \(\sigma=-1\).

We now specify the approximations used to define the weights. Use \[S=10^{30},\qquad R=3000,\qquad H=100,\qquad K=2000, \qquad W=10^9.\] For a real rational \(x\) set \[[x]=\frac{\lfloor Sx+1/2\rfloor}{S},\] and round complex numbers componentwise. Thus, if \(x=u/v\) with integers \(u,v\) and \(v>0\), its rounded numerator is \(\lfloor(2Su+v)/(2v)\rfloor\). Additions, subtractions, conjugation, and multiplication by an integer are exact unless brackets are explicitly displayed. Set \[\pi'=\frac{3141592653589793238462643383279}{S}.\] The inequality \(|\pi'-\pi|<1/S\) follows by using \(30\) terms of the alternating series for \(\arctan(1/5)\) and \(10\) for \(\arctan(1/239)\) in Machin’s identity \(\pi=16\arctan(1/5)-4\arctan(1/239)\). The total remainder bound is \[\frac{16}{61\cdot5^{61}}+\frac4{21\cdot239^{21}}.\] In particular this verification, too, uses rational arithmetic only.

The contracting and expanding roots are denoted by \(I\) and \(D\), as in the main text. For each representative center \(z\) first form \(r_z=[1/p'(z)]\), with \(p'(z)\) evaluated exactly. Store a base and a coefficient as follows: \[ (a_z,v_z)= \begin{cases} ([z],r_z),&|z|<1,\\ ([1/z],-[[1/z]r_z]),&|z|>1. \end{cases} \tag{68}\] For a contracting representative, start \(v=v_z\) and add \(2\Re v\) to \(c_0(-1)\), then to \(c_0(-2),c_0(-3),\ldots\), replacing \(v\leftarrow[va_z]\) after each addition. For an expanding representative do the same on \(c_0(0),c_0(1),\ldots\); the multiplier is \(1\) for the real root and \(2\) for the nonreal representative. All arrays start at zero. Compute \(c_0(m)\) on \[-R-K-5\le m\le H+K+5\] and define, wherever these three entries are present, \[ c_1(m)=c_0(m+3)-c_0(m)+c_0(m-1). \tag{69}\] These extra endpoint entries ensure that every subsequent use of \(c_1\) lies in its defined range. The arrays approximate the sequences \(C_0,C_1\) in the main text.

For a rational input \(c\) let \(g(c),h(c)\) be the following rounded cosine and sine values. The displayed loop fixes the order of the summation.

x = [pi' c];  y = [x*x]
t = 1;  u = x;  g = t;  h = u
for j = 1,...,50:
    t = [-t*y/(2*j*(2*j-1))]
    u = [-u*y/(2*j*(2*j+1))]
    g = g+t;  h = h+u
return (g,h)

Write \(g_A(m)=g(c_A(m))\) and \(h_A(m)=h(c_A(m))\).

The only expanding root close to the unit circle is \(\alpha\) and its conjugate. To treat their long tail, set \[ a=1/\alpha,\qquad u_0=-a/p'(\alpha),\qquad u_1=u_0(a^3-1+1/a). \tag{70}\] Their contribution to \(C_A(m)\) for \(m\ge H\) is \(u_Aa^m+\overline{u_Aa^m}\). Let \((a',u'_0)\) be the pair in (68) belonging to \(z_\alpha\). Compute \[a'_0=1,\qquad a'_{n+1}=[a'a'_n]\quad(0\le n<K),\qquad u'_1=[u'_0(a'_3-1+[1/a'])].\] The subscript in \(a'_n\) is a power-array index.

For a single moment use \(w=u'_A\) and \(k=0\). For a paired moment of types \(A,B\in\{0,1\}\), sign \(\sigma\in\{1,-1\}\), and lag \(0\le k\le K\), use \(w=u'_A+\sigma[u'_Ba'_k]\). In either case the following calculation returns an approximate moment. It retains the finite factors \(-R-k\le m<H\), omits the rapidly convergent negative tail, and treats the positive tail by summing the degree-\(2\), \(4\), and \(6\) terms of \(-\log\cos\) through geometric series. Their sum is the variable pen; the divisors \(2,12,45\) are the corresponding Taylor denominators. To compute \(e^{-\texttt{pen}}\), the algorithm first approximates \(e^{-\texttt{pen}/32}\) and then squares five times. This tail value is computed first, and the finite factors are then multiplied in increasing order of \(m\). Section 7.3 bounds all omissions and rounding errors.

U = [w*a'_H]
q_0 = 1;  q_i = [q_{i-1}*U]       for i = 1,...,6
b_0 = 1;  b_i = [b_{i-1}*pi']     for i = 1,...,6
pen = 0
for d = 2,4,6:
    mom = 0
    for j = 0,...,d:
        num = [q_j*conj(q_{d-j})]
        den = 1-[a'_j*conj(a'_{d-j})]
        mom = mom+binomial(d,j)*Re([num/den])
    pen = pen+[b_d*mom/f_d]       (f_2,f_4,f_6) = (2,12,45)
x = [-pen/32];  t = 1;  value = t
for j = 1,...,40:
    t = [t*x/j];  value = value+t
repeat five times: value = [value*value]
for m = -R-k,...,H-1, in increasing order:
    v = g_A(m)                               (single)
    v = [g_A(m)*g_B(m+k)-sigma*h_A(m)*h_B(m+k)] (pair)
    value = [value*v]
return value

Denote the outputs by \(\widetilde P_A\) and \(\widetilde P_{AB}^{\sigma}(k)\), respectively.

We use only lags \(30\le k\le2000\). With \(s=0\) for \(\sigma=1\) and \(s=1\) for \(\sigma=-1\), put \(r=\widetilde P_{00}^{\sigma}(k)\) and set \[ w_{ks}=\frac{\operatorname{sgn}(r)}{W} \left\lfloor .68W\max(0,|r|-.00026)+\frac12\right\rfloor, \qquad 30\le k\le2000. \tag{71}\] Take \(\operatorname{sgn}(0)=0\) and omit zero weights from subsequent sums. The recurrences above and this formula define the weights without any choices or numerical tolerances.

A uniform error bound for the moment calculation

We prove \[ |\widetilde P_A-P_A|<\Delta, \qquad |\widetilde P_{AB}^{\sigma}(k)-P_{AB}^{\sigma}(k)|<\Delta, \qquad \Delta=5\cdot10^{-6}, \tag{72}\] for all types, signs, and lags used above. The important issue is the weakly contracting base \(a=1/\alpha\): truncating its powers at \(H=100\) would not be accurate. The geometric sums in the tail calculation retain all of these powers.

Correlation, variance, and the absolute Gram sum

For the rest of the appendix, abbreviate \(Z_i=\mathcal Z_i^{(0)}\) and \(G_i=\mathcal G_i\) from (44) and (46). All expectations are under the stationary law of the two-sided independent bits.

The preceding calculation defines exact rational weights and bounds each moment error uniformly. We next sum those moments to prove the first three estimates of Proposition 19. A final finite-product estimate will prove the fourth.

Write \(X=-.001\), \(Y=.0055\), and let \(\mathcal R\) be the nonzero rows \((w_{ks},k,s)\) with \(30\le k\le2000\). Define \[b_A=\widetilde P_A,\qquad m_{k0}=\widetilde P_{00}^{+}(k),\quad m_{k1}=\widetilde P_{00}^{-}(k),\qquad e=\widetilde P_{01}^{+}(33).\] The approximated correlation is \[ \widetilde c=Xb_0+Ye+\sum_{(w,k,s)\in\mathcal R}wm_{ks}. \tag{78}\] For each lag \(k\), let \(J_{k0}\) and \(J_{k1}\) be respectively the second (minus) and first (plus) entries of the following pair: \[\begin{cases} (\widetilde P_{10}^{+}(k-33),\widetilde P_{10}^{-}(k-33)),&k\ge33,\\ (\widetilde P_{01}^{+}(33-k),\widetilde P_{01}^{-}(33-k)),&k<33. \end{cases}\] The approximated squared norm and absolute Gram sum are \[\begin{align*} \widetilde v={}&X^2+Y^2+2XYb_1 +2\sum_{(w,k,s)\in\mathcal R}w(Xb_0+YJ_{ks}) \\[-2pt] &+\sum_{(w,k,s),(v,l,t)\in\mathcal R} wv\,m_{|k-l|,\,\mathbf1_{s=t}},\tag{79}\\ \widetilde T={}& \sum_{\substack{(w,k,s),(v,l,t)\in\mathcal R\\k,l\ge50}} |wv\,m_{|k-l|,\,\mathbf1_{s=t}}|. \tag{80}\end{align*}\] Both double sums are over ordered pairs. Equal signs require the minus moment in a conjugated phase pair; opposite signs require the plus moment. This explains the index \(\mathbf1_{s=t}\), including the diagonal cases.

The exact finite calculations give \[ \begin{aligned} |\mathcal R|&=753,\\ |X|+|Y|+\sum_{\mathcal R}|w|&=198442946/W,\\ |X|+|Y|+\sum_{\substack{(w,k,s)\in\mathcal R\\k<50}}|w| &=11816095/W,\\ WS\widetilde c&=211815537786494774284357596751188225,\\ W^2S\widetilde v&=143920884250294630908494564120361905955993456,\\ W^2S\widetilde T&=105303788239450118397416374596041413014338334. \end{aligned} \tag{81}\] These are integer equalities: compute the arrays and products in Section 7.2, round the weights by (71), and use the unrounded rational sums (78)–(80). In particular \[211810<10^9\widetilde c<211820,\quad 143910<10^9\widetilde v<143930,\quad 105300<10^9\widetilde T<105310.\] This gives \(|G_i|<.199\) and \(|G_i^{(N)}(x)|<.199\) by the triangle inequality. The total correlation error is at most \(\Delta\cdot.199\), and the squared-norm error is at most \(\Delta\cdot.199^2\). Thus \[\mathbb E\Re(Z_iG_i)>.0002108205>.000210, \qquad \mathbb E|G_i|^2<.000144119<.000145.\]

To prepare for the last estimate, call the constant row, the type-\(1\) row, and the rows with \(k<50\) the small rows; the remaining rows are the big rows. A row of weight \(w_v\) has phase \(U_v=\exp(\pi\mathrm i\sum_m f_v(m)e_m)\), where \(e_m=2b_{i+m}-1\). Its sequence \(f_v\) is zero for the constant, \(C_1(m+33)\) for the type-\(1\) row, and \((1-2s)C_0(m+k)\) for a row \((w,k,s)\). Define the true absolute Gram sum \[ T=\sum_{v,v'\ {\rm big}}|w_vw_{v'}\mathbb E U_v\overline{U_{v'}}|. \tag{82}\] The error between this sum and (80) is at most \(\Delta\cdot.199^2\), by \(||x|-|y||\le|x-y|\). Hence \[ T<.000116,\qquad \sqrt T<.011. \tag{83}\] The total absolute weight of the small rows is less than \(.012\).

Finite products controlling the remaining moment

We now bound \(\mathbb E(Z_i^2G_i^2)\). The big rows have small coefficients on the \(71\) positions \(-70\le m\le0\). More precisely, with \(t=.06\), (73) and (74) give \[ |\tan(\pi f_v(m))|\le t \quad(v\hbox{ big},\ -70\le m\le0). \tag{84}\] Indeed \(m+k\ge-20\) for these rows, and the coefficient bound \(.018\), together with \(\sin u\le u\) and \(\cos u\ge1-u^2/2\), is sufficient.

For any real sequence \(V\) put \(c_m=|\cos(\pi V(m))|\) and \(s_m=|\sin(\pi V(m))|\). The finite products needed below are \[\begin{align*} B_2(V)&=\prod_{m=-70}^0 \min\left(1,\frac{(1+t^2)c_m+2ts_m}{1-t^2}\right),\\ B_1(V)&=\prod_{m=-70}^0 \min\left(1,\frac{c_m+ts_m}{\sqrt{1-t^2}}\right),\tag{85}\\ B_0(V)&=\prod_{m=-200}^0c_m. \end{align*}\] Their bounds are \[ B_2(2C_0)<.011,\qquad \max_{f\ {\rm small}}B_1(2C_0+f)<.015,\qquad \max_{f,g\ {\rm small}}B_0(2C_0+f+g)<.0032. \tag{86}\] Here the maxima range over the actual phase sequences of the small rows. Besides the constant and the type-\(1\) row, their eight nonzero rows are as follows: \[\begin{array}{c|rrrrrrrr} k&32&36&37&39&40&43&44&47\\ s&0&1&0&1&0&0&1&1\\ Ww_{ks}&-536932&1265024&331701&47119&1483338&436861&458216&756904 \end{array}\]

For completeness, the following upper-rounding calculation verifies all of (86). For each of its three cases and each indicated choice of small rows, form the approximate sequence \(V'\) by replacing \(C_A\) by \(c_A\). Initialize \(v=1\). For increasing \(m\) in the relevant interval, compute \[\begin{align*} b&=|g(V'(m))|+10^{-15},& y&=|h(V'(m))|+10^{-15},\\ F_0&=b,& F_1&=S^{-1}+[1.002(b+.06y)],\\ &&F_2&=S^{-1}+[1.004(1.0036b+.12y)],\\ v&\leftarrow S^{-1}+[v\min(1,F_j)]&&\text{in case }j. \end{align*}\] The factors \(1.002\) and \(1.004\) bound \((1-t^2)^{-1/2}\) and \((1-t^2)^{-1}\), respectively. The added \(10^{-15}\) exceeds the trigonometric error proved above, and each added \(S^{-1}\) compensates for a possibly downward rounding. Thus this calculation gives upper bounds, including for \(B_0\) because its true factors are at most \(1\). The maximum final numerators at scale \(S\), in the order \(B_2,B_1,B_0\), are \[ \begin{gathered} 10861284406435197401851134503,\\ 14731866010915969694738277129,\\ 3155364376927094122833206055. \end{gathered} \tag{87}\] Multiplication by \(10^6/S\) bounds them respectively by \(10862\), \(14732\), and \(3156\), proving (86).

From the finite products to the anisotropy bound

The estimates just obtained concern individual coefficients and finite products. We finish by showing how they control the full sum of phases without expanding every three-phase moment. Write \(G_i=G_{\rm small}+G_{\rm big}\) according to the preceding partition, and fix any subset \(J\subset\{-70,\ldots,0\}\). Expanding the big phases in the independent signs \(e_m\) on \(J\) gives \[ G_{\rm big}=\sum_{A\subset J}\mathrm i^{|A|}e_AH_A, \qquad e_A=\prod_{m\in A}e_m, \tag{88}\] where \[H_A=\sum_{v\ {\rm big}}w_vU_v^{\rm out} \prod_{m\in J}\cos(\pi f_v(m)) \prod_{m\in A}\tan(\pi f_v(m)).\] The phase \(U_v^{\rm out}\) depends only on signs outside \(J\), so each \(H_A\) is independent of the signs on \(J\).

We claim that \[ \|H_A\|_2\le\sqrt T\,t^{|A|}(1-t^2)^{-|J|/2}. \tag{89}\] For two big rows, independence gives their full Gram entry as the outside Gram entry times \[\prod_{m\in J}\cos(\pi f_v(m))\cos(\pi f_{v'}(m)) \bigl(1+\tan(\pi f_v(m))\tan(\pi f_{v'}(m))\bigr).\] By (84), each last factor is at least \(1-t^2\). In the expansion of \(\|H_A\|_2^2\), the extra tangent products have modulus at most \(t^{2|A|}\). Taking absolute values term by term therefore bounds the squared norm by \(Tt^{2|A|}(1-t^2)^{-|J|}\), proving (89).

First consider \(\mathbb E(Z_i^2G_{\rm big}^2)\) and set \(V=2C_0\). For a pair \(A,A'\subset J\) in (88), averaging the signs on \(J\) contributes a factor of modulus \(c_m\) where membership in \(A,A'\) agrees, and a factor of modulus \(s_m\) where it differs. The outside expectation is bounded by \(\|H_A\|_2\|H_{A'}\|_2\) using Cauchy–Schwarz and the unit modulus of the outside part of \(Z_i^2\). Summing over the four membership choices at each site gives \[|\mathbb E(Z_i^2G_{\rm big}^2)| \le T\prod_{m\in J}\frac{(1+t^2)c_m+2ts_m}{1-t^2}.\] Choose \(J\) to contain precisely those positions whose displayed factor is less than \(1\). This yields \[ |\mathbb E(Z_i^2G_{\rm big}^2)|\le T B_2(2C_0). \tag{90}\]

For a fixed small phase \(U\) with sequence \(f\), use the same expansion with \(V=2C_0+f\). There is now one subset \(A\); summing its two membership choices produces \(c_m+ts_m\). The outside expectation is bounded by \(\|H_A\|_2\). Optimizing \(J\) as before gives \[|\mathbb E(Z_i^2UG_{\rm big})| \le\sqrt T B_1(2C_0+f).\] For two small phases with sequences \(f,g\), independence directly bounds their full moment by \(B_0(2C_0+f+g)\): discard all the other absolute cosine factors, which are at most \(1\).

Finally sum these bounds with the absolute row weights. Using (83), (86), and the small-weight bound \(.012\), we obtain \[\begin{split} |\mathbb E(Z_i^2G_i^2)| &\le T\cdot.011+2(.012)\sqrt T\cdot.015 +(.012)^2\cdot.0032\\ &<.000116\cdot.011+2\cdot.012\cdot.011\cdot.015 +.012^2\cdot.0032\\ &=.0000056968<.000007. \end{split}\] This proves the remaining estimate of Proposition 19.