Cosine Polynomials With Few Zeros

In a celebrated paper, Borwein, Erdelyi, Ferguson and Lockhart constructed cosine polynomials of the formf(A)(x)=(a is an element of A)Sigma cos(ax),with A subset of N, vertical bar A vertical bar=n and as few as n(5/6+o(1)) zeros in [0,2 pi], thereby disproving an old conjecture of 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) zeros.