On complex positive definite functions on Z_n vanishing on squares

TL;DR

This paper extends the Sarkozy-Furstenberg theorem to complex positive definite functions on Z_n, demonstrating the existence of non-zero values at perfect squares under certain density conditions without relying on traditional harmonic analysis methods.

Contribution

It introduces a novel approach to analyze positive definite functions on Z_n, avoiding classical Hardy-Littlewood techniques, and establishes bounds for the van der Corput property of squares.

Findings

Existence of a perfect square s<=N/2 with f(s) non-zero for functions with density above r(N)

Bound on the van der Corput property for the set of squares

Generalization of Sarkozy-Furstenberg theorem to complex positive definite functions

Abstract

We generalize the Sarkozy-Furstenberg theorem on squares in difference sets of integers, and show that, given any positive definite function f:Z_N->C with density at least r(N), where r(N)=O((\log N)^{-c}), there is a perfect square s<=N/2 such that f(s) is non-zero. We do not rely on the usual analysis of the dichotomy of randomness and periodicity of a set and iterative application of the Hardy-Littlewood method. Instead, we find a bound for the van der Corput property of the set of squares.