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Statement

Setting (p. 1). $B$ is a standard Brownian motion and, for $\delta>0$, $F(\delta)=\mathbb P\bigl(\sup_{t\ge0}\bigl\lvert\int_0^1e^{-st}\,dB_s\bigr\rvert\le\delta\bigr)$, the function of Theorem 1.1.

Theorem 1.2 (p. 2). For $\delta\in(0,1/4)$,

$$\log F(\delta)=-\frac{2}{3\pi^2}\log^3(1/\delta)+o\bigl(\log^3(1/\delta)\bigr).$$

The paper recalls that Gao, Li and Wellner had proved $\log F(\delta)\asymp-\log^3(1/\delta)$, an estimate up to constant factors (p. 1); Theorem 1.2 identifies the constant. The paper proves the quantitative form Theorem 2.7, with error $O\bigl(\log^{5/2}(1/\delta)\sqrt{\log\log(1/\delta)}\bigr)$, and says that Theorems 1.1 and 1.2 together give the abstract's asymptotic $\liminf_{n\to\infty}\log(\lVert f_n\rVert_\infty/\sqrt n)/(\log\log n)^{1/3}=-(3\pi^2/4)^{1/3}$ almost surely (p. 2).

Source. Brayden Letwin and Mehtaab Sawhney, On the maxima of Littlewood polynomials on $[-1,1]$, arXiv:2604.19294v1 (2026). Labels and pages are those of arXiv v1: the setting on p. 1, Theorem 1.2 on p. 2, the proof in Section 2 (pp. 4--13) with Appendix A (pp. 24--29). The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

The theorem follows from Theorem 2.7 (pp. 12--13), whose proof pointer gives the route.

Dependencies

Theorem 2.7.

Bears on

• Problem 524: Theorem 1.2 is the input that turns the lower envelope $\sqrt n\,F^{-1}(\log^{-1/2}n)$ of Theorem 1.1 into the explicit logarithmic order $-(3\pi^2/4)^{1/3}(\log\log n)^{1/3}$ (Lemma 5.1, p. 18); on its own it is a statement about the Gaussian process and does not mention the problem.