Enumeration of three term arithmetic progressions in fixed density sets

TL;DR

This paper develops new real algebraic geometry methods to quantitatively estimate the number of three-term arithmetic progressions in fixed density sets, advancing from mere existence to enumeration.

Contribution

It introduces novel techniques based on real algebraic geometry to obtain quantitative counts of arithmetic progressions, extending Szemerédi's theorem beyond existence.

Findings

Quantitative bounds on three-term arithmetic progressions

New algebraic geometry methods for additive combinatorics

Discussion on potential generalizations of Szemerédi's theorem

Abstract

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.