Flat Littlewood Polynomials Exist

Abstract

We show that there exist absolute constants $\Delta >\delta >0$ such that, for all $n⩾2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta \sqrt{n}⩽|P(z)|⩽\Delta \sqrt{n}$$ for all $z\in \mathbb{C}$ with $|z|=1$. This confirms a conjecture of Littlewood from 1966.

24 pages