Limits of functions on groups

TL;DR

This paper develops a limit framework for functions on groups, analogous to graph limit theory, enabling the analysis of additive combinatorics problems and identifying exact minimizers for certain configurations.

Contribution

It introduces a new metric, convergence, and limit objects for functions on groups and compact abelian groups, advancing the theoretical tools in additive combinatorics.

Findings

Established a limit theory for functions on groups.

Identified exact minimizers for densities of linear configurations of complexity 1.

Provided a foundation for future additive combinatorics research.

Abstract

Our goal is to develop a limit approach for a class of problems in additive combinatorics that is analogous to the limit theory of dense graph sequences. We introduce metric, convergence and limit objects for functions on groups and for measurable functions on compact abelian groups. As an application we find exact minimizers for densities of linear configurations of complexity $1$.