  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.
