Letwin 2026 maxima littlewood polynomials 1 1

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theorem_1_1: Letwin and Sawhney's lower envelope for the maximum on [-1,1] of a random Littlewood polynomial: with F the small-ball probability of the sup over t >= 0 of the integral of e^(-st) dB_s over [0,1], F is continuous and strictly increasing on the positive reals, and almost surely the liminf of the maximum divided by sqrt(n) F^(-1)(log^(-1/2) n) equals 1.

theorem_1_2: Letwin and Sawhney's leading constant for the small-ball probability F of the sup over t >= 0 of the integral of e^(-st) dB_s over [0,1]: for delta in (0,1/4), log F(delta) equals -(2/(3 pi^2)) log^3(1/delta) up to an error o(log^3(1/delta)), sharpening Gao, Li and Wellner's estimate up to constant factors.

theorem_2_7: Letwin and Sawhney's quantitative small-ball estimate: for delta in (0,1/4), log F(delta) equals -(2/(3 pi^2)) log^3(1/delta) with an error O(log^(5/2)(1/delta) sqrt(log log(1/delta))), where F is the small-ball probability of the sup over t >= 0 of the integral of e^(-st) dB_s over [0,1].

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Brayden Letwin, Mehtaab Sawhney, On the maxima of Littlewood polynomials on [-1,1]. arXiv preprint (2026). arXiv:2604.19294. The copy read for this card is arXiv:2604.19294v1 (21 April 2026). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2604.19294), every other right reserved.

For a random Littlewood polynomial f_n(x) = sum over 0 <= k <= n of eps_k x^k with i.i.d. Rademacher signs, the paper determines the almost sure lower envelope of ||f_n||_infinity = max over x in [-1,1] of |f_n(x)|, complementing the Salem-Zygmund upper envelope limsup ||f_n||_infinity/sqrt(n log log n) = sqrt(2) (the paper's (1.1), p. 1). Theorem 1.1 (p. 1) shows the lower envelope is governed by a small-ball probability for a Gaussian process: with F(delta) the probability that sup over t >= 0 of the absolute value of the integral from 0 to 1 of e^{-st} dB_s is at most delta, F is continuous and strictly increasing on (0, infinity) and almost surely liminf ||f_n||_infinity / (sqrt(n) F^{-1}(1/sqrt(log n))) = 1. Theorem 1.2 (p. 2) identifies the leading constant in that small-ball rate, log F(delta) = -(2/(3 pi^2)) log^3(1/delta) + o(log^3(1/delta)) for delta in (0,1/4), sharpening Gao-Li-Wellner's estimate log F(delta) ≍ -log^3(1/delta), which held up to constant factors (p. 1); the quantitative version is Theorem 2.7 (p. 12), with error O(log^{5/2}(1/delta) sqrt(log log(1/delta))). Combining the two gives the abstract's conclusion that liminf log(||f_n||_infinity/sqrt(n))/(log log n)^{1/3} = -(3 pi^2/4)^{1/3} almost surely (through Lemma 5.1, p. 18). The method sandwiches the small-ball event between two L^2 events treatable by spectral methods, and the authors say it should apply to other Gaussian processes from sufficiently smooth kernels, a direction they do not pursue (p. 2). The paper states that the lower envelope question was raised by Salem and Zygmund and reiterated by Erdős (p. 1).

Source: https://arxiv.org/abs/2604.19294.

Bears on. #524: Theorem 1.1 gives the almost sure lower envelope of the maximum on [-1,1] of a random Littlewood polynomial, and with Theorem 1.2 (through Lemma 5.1, p. 18) the almost sure liminf of log(||f_n||_infinity/sqrt(n))/(log log n)^{1/3}, which is -(3 pi^2/4)^{1/3}, for the question the paper says Erdős reiterated from Salem and Zygmund (p. 1); the paper's f_n has coefficients indexed 0 <= k <= n, the problem's sum runs over 1 <= k <= n.

Results. Labels and pages are those of v1.

• Theorem 1.1 (p. 1): almost surely liminf ||f_n||_infinity/(sqrt(n) F^{-1}(log^{-1/2} n)) = 1, with the abstract's logarithmic form through Lemma 5.1 (p. 18).
• Theorem 1.2 (p. 2): log F(delta) = -(2/(3 pi^2)) log^3(1/delta) + o(log^3(1/delta)) for delta in (0,1/4).
• Theorem 2.7 (p. 12): the same with error O(log^{5/2}(1/delta) sqrt(log log(1/delta))).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.