Counting Zeros of Cosine Polynomials: On a Problem of Littlewood

TL;DR

This paper establishes a new lower bound on the number of zeros of cosine polynomials derived from finite sets of integers, solving a longstanding conjecture and advancing understanding of Littlewood's problem.

Contribution

It provides the first unconditional lower bound on zeros of cosine polynomials related to Littlewood's problem and introduces structural results for exponential polynomials with few coefficients.

Findings

Lower bound on zeros: at least (    |A|)^{1/2-\u03b5}

Solved a conjecture of Borwein et al.

Structural characterization of exponential polynomials with few coefficients.

Abstract

We show that if $A$ is a finite set of non-negative integers then the number of zeros of the function \[ f_A(\theta) = \sum_{a \in A} \cos(a\theta), \] in $[0,2\pi]$, is at least $(\log \log \log |A|)^{1/2-\varepsilon}$. This gives the first unconditional lower bound on a problem of Littlewood, solves a conjecture of Borwein, Erd\'elyi, Ferguson and Lockhart and improves upon work of Borwein and Erd\'elyi. We also prove a result that applies to more general cosine polynomials with "few" distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if $f$ is an exponential polynomial with "few" distinct integer coefficients and $f$ "correlates" with a low-degree exponential polynomial $P$, then $f$ has a very particular structure.