Cook 2021 universality minimum modulus random trigonometric polynomials

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Nicholas A. Cook and Hoi H. Nguyen, Universality of the minimum modulus for random trigonometric polynomials, Discrete Analysis 2021:20, 46 pp.; DOI 10.19086/da.28985; arXiv:2101.07203. Received 5 February 2021, published 6 October 2021.

The retained folder-name PDF is the journal's copy: the arXiv posting stamped "arXiv:2101.07203v3 [math.PR] 5 Oct 2021" carrying the Discrete Analysis header, article number and DOI (46 pages, pdfTeX, clean text layer; the journal is an arXiv overlay, so this posting is the published version). Provenance: retained from the repository's survey download set of September 2026; the stamp identifies the file as https://arxiv.org/abs/2101.07203v3, and the download itself was not recorded; 543,836 bytes. Earlier arXiv versions were not compared. The arXiv record (https://arxiv.org/abs/2101.07203, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Reading depth is claims checked for Theorem 1.1, Theorem 1.2 and Corollary 1.5 (pp. 2--3), read clause by clause in the text layer together with the introduction's account of Littlewood's question and the earlier bounds (pp. 1--2). Sections 2--10 (pp. 7--40) were not read.

Contents

• Setting (pp. 1--2): the Kac polynomial $F_n(z)=\sum_{j=0}^n\xi_jz^j$ with iid coefficients. By the paper's account, Littlewood ([Lit66]) asked, for Rademacher signs $\xi_j=\pm1$, whether $\min_{|z|=1}|F_n(z)|=o(1)$, and Kashin [Kas87] answered affirmatively; Konyagin's 1994 introduction instead gives Littlewood's conjecture as $\min<\varepsilon\sqrt n$ with probability tending to one, which Kashin proved. Konyagin [Kon94] showed $\mathbf P(\min_{|z|=1}|F_n(z)|\ge n^{-1/2+\varepsilon})\to0$ for every $\varepsilon>0$ (1.2); Konyagin and Schlag [KS99] showed $\limsup_n\mathbf P(\min_{|z|=1}|F_n(z)|\le\varepsilon n^{-1/2})\le C\varepsilon$ (1.3). The paper works with the normalized series $P_n(x)=(2n+1)^{-1/2}\sum_{j=-n}^n\xi_je(jx)$, which up to a unimodular factor is $(2n+1)^{-1/2}F_{2n}$ on the unit circle, and with $m_n=\min_x|P_n(x)|$ (1.4)--(1.5).
• Theorem 1.1 (p. 2, quoted from Yakir and Zeitouni [YZ]): for standard real or complex Gaussian $\xi$ and every $\tau>0$, $\lim_n\mathbf P(m_n>\tau/n)=e^{-\lambda\tau}$ with $\lambda=2\sqrt{\pi/3}$ (1.6).
• Theorem 1.2, the main result (p. 3): if $\xi$ is a centered sub-Gaussian variable of unit variance, real-valued or of the form $2^{-1/2}(\xi'+\sqrt{-1}\,\xi'')$ with iid real $\xi',\xi''$, then for every $\tau>0$, $\mathbf P(m_n>\tau/n)-\mathbf P_{N_{\mathbb R}(0,1)}(m_n>\tau/n)\to0$ (1.8). Remark 1.4 asserts, without proof, that a finite moment of sufficiently large order would suffice in place of sub-Gaussianity.
• Corollary 1.5 (p. 3): the limit (1.6) holds for every sub-Gaussian $\xi$ of mean zero and unit variance, in particular for Rademacher polynomials.
• Method (pp. 5--6, read for the plan only): the joint distribution of small values of $P_n$ at $m$ fixed points is related to a random walk in a $4m$-dimensional phase space, with small-ball estimates and a local central limit theorem under Diophantine conditions on the angles.

Compiled scope

The introduction and the statements of Section 1 were read; the proofs (Sections 2--10), the extensions of Section 10 and the bibliography were not read beyond a search for the cited works. Nothing here is independently reviewed.

Bears on. #525, as the sharpest known answer to its second question: by Corollary 1.5, for $\pm1$ coefficients and degree $2n$ the minimum modulus is $(2n+1)^{1/2}m_n$ with $n\,m_n$ converging in law to an exponential distribution of rate $\lambda$, so $m(f)$ is of order $(\deg f)^{-1/2}$ for typical $f$, and $m(f)<1$ for all but $o(2^{2n})$ sign choices, which answers the first question in even degree. The introduction (p. 2) states Littlewood's question as $\min_{|z|=1}|F_n(z)|=o(1)$ and credits Kashin with its affirmative answer; Konyagin's 1994 introduction (p. 80) gives Kashin's theorem as the bound $n^{1/2}(\log n)^{-1/3}$, which settles neither question (konyagin_1994_minimum_modulus_random_trigonometric_polynomials_coefficients). The printed rate $\lambda=2\sqrt{\pi/3}$ (Theorem 1.1, quoted from Yakir and Zeitouni, and Corollary 1.5) is twice the constant that the formal-conjectures and lean-proofs statements of the problem give, $\sqrt{\pi/12}$ in the degree normalization, i.e. $\lambda=\sqrt{\pi/3}$; the discrepancy is unresolved.