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Statement

Setting (pp. 1, 4). $F$ is the small-ball function of Theorem 1.1, $F(\delta)=\mathbb P(\sup_{t\ge0}\lvert Y_t\rvert\le\delta)$ with $Y_t=\int_0^1e^{-ut}\,dB_u$ (the paper's (2.1), p. 4).

Theorem 2.7 (p. 12). For $\delta\in(0,1/4)$,

$$\log F(\delta)=-\frac{2}{3\pi^2}\log^3(1/\delta)+O\Bigl(\log^{5/2}(1/\delta)\sqrt{\log\log(1/\delta)}\Bigr).$$

The paper presents it as the quantitative version of Theorem 1.2 (pp. 2, 12).

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: Theorem 2.7 on p. 12, its proof on pp. 12--13, the reductions it uses in Section 2 (pp. 4--12), Lemma 2.6 on p. 12 with its proof in 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

Pages 4--13 and 24--29. The substitution $Z_t=e^{t/2}Y_{e^t}$ and its stationary counterpart $X_t$, with covariance $\tfrac12\operatorname{sech}((s-t)/2)$ (the paper's (2.2)--(2.3), p. 4), reduce $F$ to $G(\delta)=\mathbb P(\sup_{t\ge0}e^{-t/2}\lvert X_t\rvert\le\delta)$: for $\delta\in(0,1/4)$, $\exp(-O(\log^2(1/\delta)))G(\delta)\le F(\delta)$ and $F(\delta/2)G(1)\exp(-O(\log^2(1/\delta)\log\log(1/\delta)))\ll G(\delta)$ (Lemma 2.1, p. 4). The $L^\infty$ event for $G$ is then compared with the $L^2$ event $H(\delta)=\mathbb P(\int_0^\infty e^{-t}X_t^2\,dt\le\delta^2)$ in both directions: by the Gaussian correlation inequality, $G(\delta)\le2H(4\delta\log(1/\delta))$ (Lemma 2.3, p. 8), and by a local comparison with a smooth cutoff, $H(\delta)\exp(-O(\log^2(1/\delta)))\le G(C\delta\log^4(1/\delta))$ (Lemmas 2.4 and 2.5, pp. 9--10). The $L^2$ asymptotic $\log\mathbb P(\int_0^\infty e^{-t}X_t^2\,dt<\delta)=-\frac1{12\pi^2}\log^3(1/\delta)+O(\log^{5/2}(1/\delta)\sqrt{\log\log(1/\delta)})$ (Lemma 2.6, p. 12), which the paper attributes in its $o(\cdot)$ form to Nazarov and Petrova's survey, is proved in Appendix A by counting the eigenvalues of the covariance operator. Since the two comparison scales differ from $\delta$ by polylogarithmic factors, the cubic main term carries over.

Dependencies

Royen's Gaussian correlation inequality; Anderson's inequality; Šidák's lemma (Lemma 2.1); Laptev's block decomposition and Karnik, Romberg and Davenport's eigenvalue bounds for prolate spheroidal wave functions (Appendix A).

Bears on

• Problem 524: it implies Theorem 1.2, which the paper uses (Lemma 5.1, p. 18) to fix the constant $(3\pi^2/4)^{1/3}$ in the logarithmic order of the lower envelope of Theorem 1.1; it does not mention the problem.