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We show that if $A$ is a finite set of non-negative integers then the numberof 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 givesthe first unconditional lower bound on a problem of Littlewood, solves aconjecture of Borwein, Erd\'elyi, Ferguson and Lockhart and improves upon workof Borwein and Erd\'elyi. We also prove a result that applies to more generalcosine polynomials with "few" distinct rational coefficients. One of the mainingredients 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 avery particular structure.

Counting Zeros of Cosine Polynomials: On a Problem of Littlewood