Infinite linear patterns in sets of positive density

TL;DR

This paper characterizes all infinite linear patterns in sets of positive density, generalizing classical results like Szemerédi's theorem and recent advances in density finite sums, revealing the structure of such configurations.

Contribution

It provides a complete description of infinite linear configurations in positive density sets, extending fundamental theorems in additive combinatorics.

Findings

All infinite linear configurations in positive density sets are characterized.

Generalizes Szemerédi's theorem to broader linear patterns.

Connects classical and recent results in additive number theory.

Abstract

In this article we describe all possible infinite linear configurations that can be found in a shift of any set of positive upper Banach density. This simultaneously generalizes Szemer\'edi's theorem on arithmetic progressions and the recent density finite sums theorem of Kra, Moreira, Richter, and Robertson.