Flat Littlewood polynomials exist

Analysis 1 Minute

You need to know: Polynomials, degree of a polynomial, set {\mathbb C} of complex numbers, absolute value |z| of complex number z.

Background: A polynomial P(z) of degree n in complex variable z is called a Littlewood polynomial if P(z) = \sum_{k=0}^n \epsilon_k z^k, where \epsilon_k \in \{-1,1\} for all 0\leq k \leq n.

The Theorem: On 22nd July 2019, Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe and Marius Tiba submitted to arxiv and Annals of Mathematics a paper in which they proved the existence of constants \Delta>\delta>0 such that, for all n\geq 2,
there exists a Littlewood polynomial P(z) of degree n with \delta\sqrt{n} \leq |P(z)| \leq \Delta\sqrt{n} for all z \in {\mathbb C} with |z|=1.

Short context: Polynomials satisfying the condition of the Theorem are called flat polynomials, hence the Theorem states that flat Littlewood polynomials exist. It answers a question of Erdos from 1957, and confirms a conjecture of Littlewood made in 1966.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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