A Hardy field extension of Szemeredi's Theorem

TL;DR

This paper extends Szemerédi's theorem by showing that arithmetic progressions can have common differences of various non-polynomial forms, including those from Hardy fields, using ergodic theory and equidistribution techniques.

Contribution

It introduces new classes of common differences for arithmetic progressions in Szemerédi's theorem, specifically those derived from Hardy field functions with certain growth conditions.

Findings

Arithmetic progressions can have common differences of the form [n^δ] for any positive δ.

Common differences can be of the form [a(n)] where a(n) belongs to a Hardy field with specific growth properties.

The proof integrates structural results for Hardy sequences, ergodic theory, and equidistribution on nilmanifolds.

Abstract

In 1975 Szemer\'edi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form $p(n)$ where $p(n)$ is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemer\'edi's theorem can be of the form $[n^\delta]$ where $\delta$ is any positive real number and $[x]$ denotes the integer part of $x$. More generally, the common difference can be of the form $[a(n)]$ where $a(x)$ is any function that is a member of a Hardy field and satisfies $a(x)/x^k\to \infty$ and $a(x)/x^{k+1}\to 0$ for some non-negative integer $k$. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.