Cosine polynomials with few zeros

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 $A\subseteq \mathbb{N}$, $\mathrm{\mid }A\mathrm{\mid }=n$ and as few as $n^{5\mathrm{/}6+o(1)}$ 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 $C(n\log n{)}^{2\mathrm{/}3}$ roots.