Cosine polynomials with few zeros

Abstract

In a celebrated paper, Borwein, Erdélyi, 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.

17 pages. A few typos fixed

References in corpus (1)

• A multivariate Berry--Esseen theorem with explicit constants