Borichev–Sodin–Weiss: Spectra of stationary processes on Z

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lemma_3: For sequences of bounded polynomially weighted norm whose spectrum lies in a fixed open arc with proper closure, one fixed linear combination of the terms at 0, ..., n-1 predicts the term at n to within any prescribed delta.

theorem_1: For a zero-mean wide-sense stationary process on Z, almost every realization has its distributional spectrum inside the support of the spectral measure, and an open proper subset of the circle that almost surely misses the realization spectra misses that support; Corollary 2 gives equality for square-integrable ergodic processes.

theorem_10: The paper's restatement of Nazarov's complete Turán lemma: the L2 norm on the circle of a trigonometric polynomial with n+1 frequencies is at most exp(A n m(T minus E)) times its L2 norm on any set E of measure at least one third, with a numerical constant A.

theorem_11: A unimodular wide-sense stationary process on Z whose spectral measure is supported in an arc of length less than pi has almost every realization of the form t s^n with random t and s on the circle; Corollary 12 identifies the ergodic such processes.

theorem_3: The paper's main theorem: a wide-sense stationary process on Z with values in a finite subset of the complex plane, whose spectral measure does not have the whole circle as support, is periodic.

theorem_4: A sequence on Z with values in a finite set X whose spectrum is not the whole circle is N-periodic, with N depending only on X and the spectrum and non-decreasing in the spectrum; the proof bounds N by a power of the size of X.

theorem_5: A stationary process on Z with values in a uniformly discrete set, whose spectral measure satisfies condition (Θ) that the squared L2 distances of 1 from polynomials vanishing at 0 are summable, has almost surely periodic realizations; Corollary 6 makes the period non-random for ergodic processes.

theorem_7: A stationary integer-valued process on Z whose spectral measure does not have the whole circle as support is periodic; the paper takes the result from Borichev, Nishry and Sodin and proves it from Theorem 5 and cyclotomic factorization.

theorem_8: For a positive measure on the circle and beta > 0, integrability of exp(beta/|theta|) gives e_n at most of order n^(-K beta) with a numerical K, while a lower density exp(-beta/|theta|) gives e_n at least of order n^(-beta/(2 pi)); so a deep exponential zero forces condition (Θ), and a shallow one can violate it.

theorem_9: For a weight W at least 1 and its truncation W_A = min(W, e^A), an integrability condition on W^beta against rho and a Poisson-smoothing condition on log W_A bound e_n(rho) above by exp of minus a multiple of the integral of log W_A, and a lower density W^(-beta) bounds it below.

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The copy read for this card is the arXiv preprint, version 1 of arXiv:1701.03407 (12 January 2017, 22 pages); the labels and pages cited on this card and its result pages are that version's. The arXiv record names arXiv's non-exclusive distribution license, every other right reserved.

Alexander Borichev, Mikhail Sodin, Benjamin Weiss, "Spectra of stationary processes on Z," arXiv:1701.03407 (2017).

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Bears on. E1150 (context only): the paper does not mention the problem, and none of its results bounds the maximum modulus of a polynomial with coefficients $\pm1$. Theorems 3, 4, 5 and 11, Lemma 3 and Theorem 10 enter the stationary-limit discussion under Relation to E1150 below (Theorem 1 only through the proofs of Theorems 3 and 11). That discussion finds that the spectral measure of a stationary limit of $\pm1$ polynomials of maximum modulus $(1+o(1))\sqrt n$ is Lebesgue measure, outside the hypotheses of Theorems 3, 5 and 11.

Overview

The paper asks how the spectral measure $\rho$ of a stationary process on $\mathbb Z$ is constrained when the process takes values in a finite, discrete, or unimodular set. Its principal conclusion is a rigidity theorem: a finite-valued wide-sense stationary process whose spectral support is a proper subset of $\mathbb T$ must be periodic (Theorem 3, §3). Here $r(m)=\mathbb E[\xi(0)\overline{\xi(m)}]=\widehat\rho(m)$, and the spectrum of the process is ${\rm spt}(\rho)$.

The bridge from process spectra to individual realizations is Theorem 1 (§2.2). For a zero-mean wide-sense stationary process, the distributional spectrum $\sigma(\xi)$ of almost every realization is contained in ${\rm spt}(\rho)$; conversely, any fixed proper open set that almost surely misses $\sigma(\xi)$ also misses ${\rm spt}(\rho)$. The proof uses the isometry sending $\xi(n)$ to $t^n$ between the closed linear span of the process in $L^2(\mathbb P)$ and $L^2(\rho)$: vanishing of the integral in equation (1) is equivalent to the almost-sure linear identity (2). Corollary 2 gives $\sigma(\xi)={\rm spt}(\rho)$ almost surely for square-integrable ergodic processes. Sections 2.3–2.4 recall, as cited background, that for polynomially growing sequences this distributional spectrum agrees with the Carleman spectrum, and for bounded sequences with the Beurling spectrum.

The deterministic ingredient is Theorem 4 (§3.1), a strengthened form of Helson's theorem: if $X\subset\mathbb C$ is finite and $\xi:\mathbb Z\to X$ has $\sigma(\xi)\ne\mathbb T$, then $\xi$ is periodic, with a period depending only on $X$ and $\sigma(\xi)$. Lemma 1 (§3.2), proved by Runge approximation, constructs a polynomial equal to $1$ at zero and uniformly small on a proper closed arc; Lemma 2 is the cited Videnskii Bernstein inequality on an arc. These feed into the $\delta$-prediction Lemma 3 (§3.3): for sequences with bounded weighted norm and spectrum in a fixed open arc whose closure is not all of $\mathbb T$, one future term is approximated, to prescribed accuracy, by a fixed linear combination of a finite preceding block. For finite $X$, choosing the error below half the minimum spacing makes the prediction exact. Applying the same argument to the reversed sequence shows that one finite block determines the entire sequence; the pigeonhole argument in §3.4 then gives a period $N\le |X|^q$, where $q$ is the prediction length supplied by Lemma 3. Theorem 3 follows by combining this result with Theorem 1.

Section 4 replaces a literal spectral gap by an approximation condition. With $e_n(\rho)={\rm dist}_{L^2(\rho)}(1,\mathcal P_n^0)$, condition $(\Theta)$ is $\sum_{n\ge1}e_n(\rho)^2<\infty$ (§4.1). Theorem 5 (§4.2) states that a stationary process taking values in a uniformly discrete set has almost surely periodic realizations whenever $\rho$ satisfies $(\Theta)$; Corollary 6 makes the period non-random in the ergodic case. The proof in §4.3 turns the extremal polynomial defining $e_N(\rho)$ into a probabilistic predictor: the failure probability is at most $4e_N(\rho)^2/\delta_X^2$, and the two-sided reconstruction failure is bounded by $8\delta_X^{-2}\sum_{n\ge N}e_n(\rho)^2$. Theorem 7 (§§4.4–4.5), which the paper takes from Borichev–Nishry–Sodin (its reference [2]), separately treats integer-valued stationary processes with proper spectral support and obtains a common period; the proof given here applies Theorem 5 and then uses rational generating functions and factorization of their denominators into cyclotomic polynomials.

Theorems 8 and 9 (§5) quantify when $(\Theta)$ can hold near an exponential zero of $\rho$. Theorem 8(A) gives $e_n(\rho)\lesssim n^{-K\beta}$ under $\int \exp(\beta/|\theta|)\,d\rho(e^{i\theta})<\infty$, while Theorem 8(B) gives the converse-type lower estimate $e_n(\rho)\gtrsim_\beta n^{-\beta/(2\pi)}$ when $d\rho\gtrsim \exp(-\beta/|\theta|)\,dm$. Thus part (A) implies $(\Theta)$ when the resulting exponent satisfies $2K\beta>1$, not for every $\beta$. The more general Theorem 9 derives upper and lower bounds through truncated weights $W_A$, assumptions (3)–(4), outer functions (§5.1), and the cited Nazarov Turán inequality restated as Theorem 10 (§5.2); equation (5) verifies the required Poisson estimate for $W(e^{i\theta})=\exp(1/|\theta|)$. Theorem 9(A) is stated for $n\ge A^2\beta/(2M)$, while its printed proof (p. 15) takes $n\ge A^2\beta/M$; the paper does not comment on the difference.

Finally, the paper distinguishes finite-valued rigidity from the general unimodular case. Section 6.1 constructs a unimodular stationary process with any prescribed probability measure as spectral measure, so unimodularity alone imposes no support restriction. Under the stronger hypothesis that ${\rm spt}(\rho)$ lies in an arc of length less than $\pi$, Theorem 11 (§6.2) shows that almost every realization is geometric, $\xi(n)=t s^n$, equation (6). Corollary 12 classifies the ergodic case. The proof invokes the external Eremenko–Ostrovskii analytic-continuation theorem, restated as Theorem 13, after Theorem 1 identifies the required Carleman continuation domain.

Relation to E1150

Write an E1150 polynomial as

$$P_n(z)=\sum_{j=0}^{n}\varepsilon_jz^j,\qquad \varepsilon_j\in\{-1,1\},$$

and put $N=n+1$. The paper does not prove a lower bound of the form $\|P_n\|_{L^\infty(\mathbb T)}>(1+c)\sqrt n$, and none of its numbered results supplies such a constant. Its objects are bi-infinite stationary processes and their spectral measures, whereas E1150 concerns every individual finite coefficient word.

There is nevertheless an exact stationary translation. Extend $(\varepsilon_0,\ldots,\varepsilon_{N-1})$ periodically and let

$$\xi_N(k)=\varepsilon_{U+k\bmod N},$$

where $U$ is uniform on $\mathbb Z/N\mathbb Z$. This is a stationary $\{-1,1\}$-valued process of period $N$. If $\omega=e^{2\pi i/N}$, its spectral measure, in the paper's Fourier convention, is

$$\rho_N=\frac1{N^2}\sum_{\ell=0}^{N-1}|P_n(\omega^\ell)|^2\,\delta_{\omega^\ell}.$$

Thus values of the Littlewood polynomial at the $N$-th roots of unity are precisely the spectral masses of the randomized cyclic word. Theorem 3 applies because $\rho_N$ has finite support, but concludes only that $\xi_N$ is periodic—already built into the construction. Likewise, $e_k(\rho_N)=0$ for $k\ge N$, since $z^N=1$ on its support, so Theorem 5 again recovers only this known periodicity.

The paper is more useful as an exclusion principle in a limiting argument. If a family of Littlewood words produced a stationary $\{-1,1\}$-valued limit whose spectral measure had a fixed open gap, Theorem 3 would force that limit to be periodic. If its support lay in an arc of length less than $\pi$, Theorem 11 and equation (6) would force almost every realization to be geometric; because all values are $\pm1$, this reduces to $t,s\in\{\pm1\}$, hence a constant or alternating sequence. Lemma 3 and §3.4 could make this route quantitative when one has a fixed containing arc: a finite block determines the sequence and the period is at most $2^q$. The paper does not relate the prediction length $q$, or the size of a spectral gap, to $\|P_n\|_\infty/\sqrt n$.

In fact, the natural spectral limit of flat Littlewood polynomials, the objects a negative answer to E1150 consists of, lies outside these rigidity hypotheses. A negative answer to E1150 gives degrees $n\to\infty$ and polynomials $P_n$ with

$$\|P_n\|_\infty\le(1+o(1))\sqrt n=(1+o(1))\sqrt N.$$

Parseval gives $\int_{\mathbb T}|P_n|^2\,dm=N$. Hence $f_n=|P_n|^2/N$ has integral $1$, is bounded above by $1+o(1)$, and therefore converges to $1$ in $L^1(m)$. For each fixed $r\ge1$,

$$\widehat f_n(r)=\frac1N\sum_{j=0}^{N-1-r}\varepsilon_{j+r}\varepsilon_j\longrightarrow0.$$

Consequently, any stationary local limit obtained from random translates has correlations $r(0)=1$ and $r(k)=0$ for $k\ne0$, hence spectral measure equal to normalized Lebesgue measure $m$, with full support $\mathbb T$. For this measure $e_k(m)=1$ for every $k$, so condition $(\Theta)$ also fails. Thus Theorems 3–5 are compatible with, and do not rule out, asymptotically ultraflat Littlewood polynomials.

Theorem 10 supplies an $L^2$ concentration inequality for a polynomial with $n+1$ frequencies, but its factor is exponential in $n\,m(\mathbb T\setminus E)$; together with Parseval it does not yield the fixed strict $L^\infty$ gap required by E1150. The paper was therefore consulted for its spectral-rigidity framework and for possible stationary-limit obstructions, not for a direct resolution of the finite-polynomial extremal problem.

Results

• Theorem 1 (p. 4): realization spectra lie in the spectrum of the process, with Corollary 2
• Theorem 3 (p. 7): a finite-valued process with a spectral gap is periodic
• Theorem 4 (p. 7): Helson's theorem with a period depending only on the values and the spectrum
• Lemma 3 (p. 8): the delta-prediction lemma, with Lemmas 1 and 2
• Theorem 5 (p. 11): condition (Θ) and periodic realizations, with Corollary 6
• Theorem 7 (p. 12): integer-valued processes with a spectral gap
• Theorem 8 (p. 13): decay of e_n at an exponential zero
• Theorem 9 (p. 14): bounds for e_n through truncated weights
• Theorem 10 (p. 17): Nazarov's Turán-type lemma, as restated
• Theorem 11 (p. 20): unimodular processes with spectrum in a short arc, with Corollary 12

The result pages state the results and point to the proofs; no proof is transcribed. Read status: claims checked for the statements on those pages, read clause by clause on the pages of the version named above; Lemma 2, Theorem 10 and Theorem 13 are recorded as the paper cites them and were not checked against their own sources. Nothing is independently reviewed.

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