An improved lower bound for a problem of Littlewood on the zeros of cosine polynomials

TL;DR

This paper establishes a significantly improved lower bound on the minimum number of zeros of cosine polynomials with integer coefficients, advancing the longstanding problem posed by Littlewood regarding the zeros of such functions.

Contribution

The paper provides a new lower bound for the zeros of cosine polynomials, improving previous bounds exponentially and advancing understanding of Littlewood's problem.

Findings

Lower bound Z(N) ≥ (log log N)^{1+o(1)}

Exponential improvement over previous bounds

Progress towards solving Littlewood's problem

Abstract

Let $Z(N)$ denote the minimum number of zeros in $[0,2\pi]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $|A|=N$. It is an old problem of Littlewood to determine $Z(N)$. In this paper, we obtain the lower bound $Z(N)\geqslant (\log\log N)^{(1+o(1))}$ which exponentially improves on the previous best bounds of the form $Z(N)\geqslant (\log\log\log N)^c$ due to Erd\'elyi and Sahasrabudhe.