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Statement

Setting (p. 1). A Littlewood polynomial of degree $n$ is $f_n(x)=\sum_{k=0}^n\varepsilon_kx^k$ with $\varepsilon_k\in\{-1,1\}$; here $(\varepsilon_k)_{k\ge0}$ are independent Rademacher signs, so $f_n$ has the $n+1$ coefficients $\varepsilon_0,\dots,\varepsilon_n$, and $\lVert f_n\rVert_\infty=\max_{x\in[-1,1]}\lvert f_n(x)\rvert$.

Theorem 1.1 (p. 1). Let $B$ be a standard Brownian motion and, for $\delta>0$, let

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

Then $F$ is continuous and strictly increasing on $(0,\infty)$, so it has an inverse $F^{-1}\colon(0,1)\to(0,\infty)$, and almost surely

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

The upper envelope it complements is Salem and Zygmund's, recalled as the paper's (1.1) (p. 1): almost surely $\limsup_{n\to\infty}\lVert f_n\rVert_\infty/\sqrt{n\log\log n}=\sqrt2$.

Consequence (abstract, p. 1; Lemma 5.1, p. 18). With $b_n=F^{-1}(\log^{-1/2}n)$ and $s_n=(\log\log n)^{1/3}$ (the paper's (5.1), p. 17), Lemma 5.1 states that for all sufficiently large $n$

$$\log(1/b_n)=\Bigl(\Bigl(\frac{3\pi^2}{4}\Bigr)^{1/3}+o(1)\Bigr)s_n,$$

which it derives from Theorem 1.2. Together with Theorem 1.1 this gives the abstract's statement that almost surely

$$\liminf_{n\to\infty}\frac{\log\bigl(\max_{x\in[-1,1]}\lvert f_n(x)\rvert/\sqrt n\bigr)}{(\log\log n)^{1/3}}=-\Bigl(\frac{3\pi^2}{4}\Bigr)^{1/3}.$$

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 and Theorem 1.1 on p. 1, the proof in Sections 3--5 (pp. 13--23), Lemma 5.1 on p. 18. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the statement, Lemma 5.1 and the abstract's statement were read clause by clause on the printed pages. The proof was read but not checked step by step; the continuity and strict monotonicity of $F$ are sketched in the paper, not proved in detail (p. 15). Nothing here is independently reviewed.

Proof pointer

Pages 13--23. Writing $x=\pm e^{-t/n}$ puts logarithmic coordinates at the two endpoints, and $\lVert f_n\rVert_\infty$ is the largest of $1$ and the suprema of the two endpoint profiles (Lemma 3.1, p. 13). A Komlós--Major--Tusnády coupling of the even and odd coefficients with two independent Brownian motions (Lemmas 3.2 and 3.3, p. 14) puts both profiles within $O(\log n)$ of $\sqrt n$ times two independent copies of the process $Y_t=\int_0^1e^{-st}\,dB_s$, outside an event of probability $\ll n^{-2}$. Section 4 records that $F$ is continuous and strictly increasing, which the paper calls a routine exercise in the theory of Gaussian processes and only sketches (p. 15), and quantifies how $F$ and $F^{-1}$ change under small multiplicative perturbations (Proposition 4.1, Corollaries 4.2 and 4.3). Section 5 fixes the scale $b_n$ and its stability on dyadic blocks (Lemmas 5.1 and 5.2), proves $\liminf\ge1$ by a first-moment argument on a geometric mesh with Borel--Cantelli (Proposition 5.5, p. 20), and proves $\liminf\le1$ by splitting $f_{N_{j+1}}$ into an old part and an independent fresh block that is small infinitely often (Proposition 5.6, p. 22).

Dependencies

The Komlós--Major--Tusnády strong approximation; the small-ball asymptotic of Theorem 1.2 (through Lemma 5.1 and Section 4); the Gaussian $B$-inequality of Cordero-Erausquin, Fradelizi and Maurey (Proposition 4.1, p. 15); Salem and Zygmund's upper envelope (1.1), used to show that the old part is negligible (p. 23).

Bears on

• Problem 524: the problem asks for the order of magnitude, for almost every $t$, of the maximum on $[-1,1]$ of the $\pm1$ polynomial built from the binary digits of $t$. The paper states that determining the lower envelope of $\lVert f_n\rVert_\infty$ was raised by Salem and Zygmund and reiterated by Erdős (p. 1). Theorem 1.1 gives that lower envelope as $\sqrt n\,F^{-1}(\log^{-1/2}n)$, and with Lemma 5.1 its logarithmic order $\log(\lVert f_n\rVert_\infty/\sqrt n)\sim-(3\pi^2/4)^{1/3}(\log\log n)^{1/3}$ along the liminf; the paper's $f_n$ has coefficients indexed $0\le k\le n$, while the problem's sum runs over $1\le k\le n$.