  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.
