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Claim. For a Littlewood polynomial $f_n(x)=\sum_{k=0}^n\epsilon_kx^k$ with independent Rademacher signs, write $\lVert f_n\rVert_\infty=\max_{x\in[-1,1]}\lvert f_n(x)\rvert$. Theorem 1.1: let $B$ be a standard Brownian motion and, for $\delta>0$,

$$F(\delta)=\mathbb P\Bigl(\sup_{t\ge0}\Bigl\lvert\int_0^1e^{-st}\,dB_s\Bigr\rvert\le\delta\Bigr);$$

then $F$ is continuous and strictly increasing on $(0,\infty)$ and almost surely

$$\liminf_{n\to\infty}\frac{\lVert f_n\rVert_\infty}{\sqrt n\,F^{-1}(\log^{-1/2}n)}=1 .$$

Theorem 1.2: for $\delta\in(0,1/4)$, $\log F(\delta)=-\frac{2}{3\pi^2}\log^3(1/\delta)+o(\log^3(1/\delta))$. The two theorems give the statement of the abstract: almost surely

$$\liminf_{n\to\infty}\frac{\log\bigl(\lVert f_n\rVert_\infty/\sqrt n\bigr)}{(\log\log n)^{1/3}} =-\Bigl(\frac{3\pi^2}{4}\Bigr)^{1/3}.$$

For almost every $t$ the signs $(-1)^{\epsilon_k(t)}$ of Problem 524 are independent Rademacher variables, and the paper's sum starts at $k=0$ where the problem's starts at $k=1$, which changes the maximum by at most $1$, so the result is a statement about $M_n(t)$. The introduction recalls Salem and Zygmund's upper envelope, $\limsup_n\lVert f_n\rVert_\infty/\sqrt{n\log\log n}=\sqrt2$ almost surely [SaZy54, Theorem (6.1.1)], and presents the lower envelope as the question raised there and reiterated by Erdős in his 1961 problem paper [Er61], which is this problem's source. Together the two envelopes determine the almost sure behavior of $M_n(t)$ that the problem asks for, so the claim is recorded as answered. The proof sandwiches the small-ball event of the Gaussian process between two $L^2$ events treated by spectral methods and reduces the polynomial to the Gaussian model by a strong approximation; the paper's acknowledgments say that ChatGPT Pro drew the authors' attention to the Gaussian inequality used in Proposition 4.1 and that ChatGPT Codex was used to help write the manuscript. The source card is Letwin and Sawhney 2026.

Depends on. No page of this wiki.

Standing. Claimed. The paper is an arXiv preprint (v1 submitted 2026-04-21), not refereed and not registered on the site's proof-claims tab; Nat Sothanaphan announced it on the thread on 2026-04-24. On 2026-01-30 Sawhney had written on the thread that Sawhney and Letwin could prove a result stronger than the note on Chojecki 2026 by determining the constant in the Gao–Li–Wellner small-ball asymptotic, and that neither result pins the answer down as precisely as Salem and Zygmund might have wanted. The site labels the problem OPEN (page last edited 27 December 2025, before these postings). Nothing is compiled or reviewed in this wiki.