Counting Zeros of Cosine Polynomials: On a Problem of Littlewood

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.