On van der Corput property of squares

TL;DR

This paper establishes an upper bound for the van der Corput property of perfect squares, showing it diminishes at a rate of (log n)^{-1/3} through the construction of specialized trigonometric polynomials.

Contribution

It provides a new upper bound for the van der Corput property of squares, answering a question posed by Ruzsa and Montgomery, using a novel polynomial construction approach.

Findings

Upper bound for van der Corput property: O((log n)^{-1/3})

Construction of non-negative, normed trigonometric polynomials with spectrum in squares

Small free coefficient a_0=O((log n)^{-1/3})

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}).