On van der Corput property of squares

Abstract

We prove that the upper bound for the van der Corput property of the set of perfect squares is O((log n)^{-1/3}), giving an answer to a problem considered by Ruzsa and Montgomery. We do it by constructing non-negative valued, normed trigonometric polynomials with spectrum in the set of perfect squares not exceeding n, and a small free coefficient a_0=O((log n)^{-1/3}).