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

Abstract

A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with ${\epsilon }_k\in \{-1,1\}$. Let $({\epsilon }_k{)}_{k\ge 0}$ 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, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3π^2}{4}\Big)^{1/3}. \]

29 pages