On an Erdös Problem about the Maximum Modulus of Littlewood Polynomials on the Unit Circl

Abstract

Making a progress in an Erdös problem dating back to 1957, we show that $$\underset{t\in \mathbb{R}}{\max }|P_n(e^{it}){|}^2\ge n+1+\frac{1}{38}{n}^{1/3}$$ for every polynomial $P_n$ of degree $n$ with all of its coefficients in $\{-1,1\}$.