# Flat Littlewood polynomials exist

### Abstract

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

## Authors

Paul Balister

Mathematical Institute, University of Oxford, Oxford, UK

Béla Bollobás

Department of Pure Mathematics and Mathematical Statistics, Cambridge, UK and Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA

Robert Morris

IMPA, Rio de Janeiro, Brazil

Julian Sahasrabudhe

Peterhouse, University of Cambridge, Cambridge, UK

Marius Tiba

Department of Pure Mathematics and Mathematical Statistics, Cambridge, UK