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
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