An optimal version of Sarkozy's theorem

TL;DR

This paper proves a result about the density of a subset of integers intersecting with its quadratic shifts, using Fourier analysis, providing a clearer exposition of a special case related to Sarkozy's theorem.

Contribution

It presents a simplified proof of a special case of Sarkozy's theorem using Fourier techniques, clarifying key ideas from more complex prior work.

Findings

Existence of t with specified intersection density for large N

Fourier analysis as a tool for additive combinatorics

Simplified exposition of a complex theorem

Abstract

Using Fourier analytic techniques, we prove that if $\VE>0$, $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,...,N\}$, then there must exist $t\in\N$ such that \[\frac{|A\cap (A+t^2)|}{N}>(\frac{|A|}{N})^2-\VE.\] This is a special case of results presented in Lyall and Magyar \cite{LM3} and we will follow those arguments closely. We hope that the exposition of this special case will serve to illuminate the key ideas contained in \cite{LM3}, where many of the analogous arguments are significantly more technical.