On nonnegative cosine polynomials with nonnegative integral coefficients

Abstract

We show that there exist p 0 > 0 {p_0} > 0 and p 1 , … , p N {p_1}, \ldots ,{p_N} nonnegative integers, such that \[ 0 ⩽ p ( x ) = p 0 + p 1 cos ⁡ x + ⋯ + p N cos ⁡ N x 0 \leqslant p(x) = {p_0} + {p_1}\cos x + \cdots + {p_N}\cos Nx \] and p 0 ≪ s 1 / 3 {p_0} \ll {s^{1/3}} for s = ∑ j = 0 N p j s = \sum \nolimits _{j = 0}^N {{p_j}} , improving on a result of Odlyzko who showed the existence of such a polynomial p p that satisfies p 0 ≪ ( s log ⁡ s ) 1 / 3 {p_0} \ll {(s\log s)^{1/3}} . Our result implies an improvement of the best known estimate for a problem of Erdős and Szekeres. As our method is nonconstructive, we also give a method for constructing an infinite family of such polynomials, given one good "seed" polynomial.