An approximate logic for measures

TL;DR

This paper introduces a logical framework connecting combinatorics and measure theory, enabling formalization of key results like the Furstenberg correspondence and Szemerédi's Regularity Lemma, and explores links to stability and Gowers norms.

Contribution

It develops a new logical system that unifies combinatorial and measure-theoretic concepts, providing simplified proofs and new insights into their relationships.

Findings

Expressed the Furstenberg correspondence within the logic

Provided a short proof of Szemerédi's Regularity Lemma

Connected model-theoretic stability to Gowers uniformity norms

Abstract

We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemer\'edi Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.