On the maxima of Littlewood polynomials on $[-1,1]$

TL;DR

This paper investigates the maximum absolute value of Littlewood polynomials with random coefficients on the interval [-1,1], revealing a probabilistic limit involving Gaussian processes and logarithmic asymptotics.

Contribution

It establishes a connection between the maximum modulus of Littlewood polynomials and small-ball probabilities of Gaussian processes, providing a precise asymptotic limit.

Findings

Almost sure limit of the scaled maximum modulus is characterized.

The lower envelope is linked to small-ball probabilities of Gaussian processes.

Asymptotic behavior involves a specific constant with a (log log n)^{1/3} scaling.

Abstract

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3\pi^2}{4}\Big)^{1/3}. \]