  In a celebrated paper, Borwein, Erd\'elyi, Ferguson and Lockhart constructed
cosine polynomials of the form \[ f_A(x) = \sum_{a \in A} \cos(ax), \] with
$A\subseteq \mathbb{N}$, $|A|= n$ and as few as $n^{5/6+o(1)}$ zeros in
$[0,2\pi]$, thereby disproving an old conjecture of J.E. Littlewood. Here we
give a sharp analysis of their constructions and, as a result, prove that there
exist examples with as few as $C(n\log n)^{2/3}$ roots.
