Polynomial Configurations in Difference Sets (Revised Version)

TL;DR

This paper proves a quantitative version of the Polynomial Szemeredi Theorem for difference sets, extending previous results by establishing higher dimensional analogues and applying a lifting argument.

Contribution

It introduces a higher dimensional analogue of Sarkozy's theorem and applies a lifting technique to advance the understanding of polynomial configurations in difference sets.

Findings

Difference sets of positive density contain polynomial configurations.

Established a higher dimensional analogue of Sarkozy's theorem.

Applied a lifting argument to derive the main result.

Abstract

We prove a quantitative version of the Polynomial Szemeredi Theorem for difference sets. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sarkozy (the simplest non-trivial case of the Polynomial Szemeredi Theorem asserting that the difference set of any subset of the integers of positive upper density necessarily contains a perfect square) and then applying a simple lifting argument.