A new proof of Sarkozy's theorem

TL;DR

This paper provides a new, simplified proof of Sarkozy's theorem, which states that any subset of natural numbers with positive upper density contains two elements differing by a perfect square.

Contribution

It introduces a novel proof technique by adapting an argument from Croot and Sisask, offering a more straightforward proof of Sarkozy's theorem.

Findings

Confirmed the existence of perfect square differences in dense subsets of natural numbers

Provided a simpler proof method for Sarkozy's theorem

Enhanced understanding of additive combinatorics techniques

Abstract

It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.