On the regularity of almost stable relations

TL;DR

This paper develops a theory of local stability up to measure-zero sets, establishing a regularity lemma for graphs with almost stable relations and applications to finite groups.

Contribution

It introduces a general framework for almost stable formulas, linking model theory with combinatorics and group theory, including new regularity lemmas and stabilizer subgroup results.

Findings

Finite graph regularity lemma for almost stable relations

Existence of definable stabilizer subgroups in groups

Finite arithmetic regularity lemma for almost stable relations

Abstract

We develop a general theory of local stability up to belonging to an ideal (e.g. having measure zero). From a model-theoretic perspective, we prove a stationarity principle for almost stable formulas in this sense, and build a topological space of partial types whose Cantor-Bendixson rank is finite. The interaction of this space with Keisler measures and definable groups yields, on the one hand, a regularity lemma for infinite graphs where the edge relation is almost stable, and, on the other hand, the existence of definable stabilizer subgroups. As an application, we prove a finite graph regularity lemma and an arithmetic regularity lemma for almost stable relations in arbitrary finite groups.