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Cook and Nguyen [CoNg21] work with the normalized trigonometric polynomial $P_n(x)=(2n+1)^{-1/2}\sum_{j=-n}^n\xi_je(jx)$ with independent copies $\xi_j$ of a centered random variable $\xi$ of unit variance, and with $m_n=\min_x|P_n(x)|$. Up to a factor of modulus one, $P_n$ is the restriction to the unit circle of $(2n+1)^{-1/2}$ times the degree-$2n$ polynomial with coefficients $\xi_j$. Their Theorem 1.2 states that if $\xi$ is sub-Gaussian, real-valued or of the form $2^{-1/2}(\xi'+\sqrt{-1}\,\xi'')$ with independent, identically distributed real $\xi',\xi''$, then for every $\tau>0$ the probability $\mathbb{P}(m_n>\tau/n)$ differs from its value for standard Gaussian coefficients by $o(1)$. With Yakir and Zeitouni's Gaussian limit law, this gives Corollary 1.5: for every such $\xi$, in particular for uniform $\pm1$ signs,

$$\lim_{n\to\infty}\mathbb{P}\Bigl(m_n>\frac{\tau}{n}\Bigr)=e^{-\lambda\tau}, \qquad\lambda=2\sqrt{\pi/3}.$$

For the problem's $m(f)=\min_{|z|=1}|f(z)|$ this says that $(\deg f)^{1/2}m(f)$ converges in law to an exponential distribution with an explicit rate, which settles the second question of Problem 525, the behavior of $m(f)$, at the scale $n^{-1/2}$ whose exponent Konyagin's upper bound (Konyagin 1994) and Konyagin and Schlag's lower bound (Konyagin and Schlag 1999) had shown to be optimal. Since $m(f)$ then exceeds any fixed bound with probability tending to zero, it contains the first question's answer, first proved by Konyagin's bound; the paper's introduction credits that answer to Kashin, whose bound is weaker (Kashin 1987). The claim value is answered: the first question is answered yes and the second, which asks for the behavior of $m(f)$, is determined rather than proved or disproved. The paper states the theorem for even degree $2n$ and says, on p. 2, that "all of our arguments extend to the case of odd degree"; the odd-degree case is thus the authors' assertion, not a separately printed theorem. The method relates the joint distribution of small values of $P_n$ at finitely many points to a random walk in a phase space of dimension four times the number of points, with small-ball estimates and a local central limit theorem under Diophantine conditions on the angles; the source card digests the paper. The page's date is the first arXiv posting, 2021-01-18.

The rate constant. The paper prints $\lambda=2\sqrt{\pi/3}$ for real and complex coefficients alike: Theorem 1.1 quotes Yakir and Zeitouni's law with that constant for standard real or complex Gaussian coefficients, and Corollary 1.5 carries it to every sub-Gaussian $\xi$; Yakir and Zeitouni's own paper (arXiv:2006.08943) prints the same constant for complex Gaussian coefficients in its Theorem 1 and for real Gaussian coefficients in its Theorem 2. The formal-conjectures statement file for the problem and the lean-proofs development linked above instead state, for $\pm1$ signs in the degree normalization, that $\mathbb{P}(m(f)>\varepsilon(\deg f)^{-1/2})\to e^{-\sqrt{\pi/12},\varepsilon}$; since $m(f)=(2n+1)^{1/2}m_n$ with $\deg f=2n$, this is $\lambda=\sqrt{\pi/3}$ in the paper's normalization, half the printed rate. One reading of the halving, reviewed nowhere the corpus knows of, is that for real coefficients $P_n(-x)=\overline{P_n(x)}$, so the near-minima come in conjugate pairs and the limiting intensity of the real case is half that of the complex case. The discrepancy between the printed Theorem 1.1 and Corollary 1.5, with Yakir and Zeitouni's Theorem 2, and the two Lean statements is recorded as unresolved; the qualitative result, universality and an exponential limit law for $(\deg f)^{1/2}m(f)$, does not depend on it.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Discrete Analysis 2021, Paper No. 20, 46 pp., received 2021-02-05 and published 2021-10-06; the journal is an arXiv overlay, so the paper link and the third arXiv version are the same text. Reviewed: the site's curator, Thomas F. Bloom, records the limiting distribution as Cook and Nguyen's in the problem's commentary. The statements of Theorem 1.1, Theorem 1.2 and Corollary 1.5 are checked against the paper; the proofs are not compiled in this wiki. Formalization: a Lean 4 file in the lean-proofs repository, linked above at its pinned commit, declares itself a formalization of Cook and Nguyen's solution, names Codex and GPT-5.6 Sol as its formal authors, and proves erdos_525, the conjunction of a limit law for the proportion of degree-$N$ sign polynomials with minimum modulus above $\tau/\sqrt N$, the statement that the exceptional polynomials of the first question number $o(2^N)$, and the probability form of the latter, from even- and odd-degree developments it imports; the file ends with an axiom print. The site's page links the formal-conjectures statement file for the problem, which points to that proof. This corpus has neither built the development nor audited its statement, whose rate constant differs from the paper's as recorded above, so no formalized evidence is listed.