Erd\H{o}s Semi-groups, arithmetic progressions and Szemer\'edi's theorem

TL;DR

This paper introduces a special type of sub semi-group of the real numbers modulo 1, exploring its connection to Szemerédi's theorem on arithmetic progressions, and providing new insights into the structure of such semi-groups.

Contribution

It defines and analyzes Erdős semi-groups, revealing their relationship with Szemerédi's theorem and advancing understanding of arithmetic progressions within these structures.

Findings

Erdős semi-groups are closely related to arithmetic progressions.

The structure of these semi-groups provides new perspectives on Szemerédi's theorem.

The study uncovers properties linking semi-group theory and additive combinatorics.

Abstract

In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.