Submodular functions in additive combinatorics problems for group actions and representations

TL;DR

This paper explores the application of submodular functions to classical problems in additive combinatorics within the context of group actions and representations, introducing new theoretical insights and tools.

Contribution

It introduces the notion of left invariant submodular functions on power sets and applies them to analogues of additive combinatorics problems for groups.

Findings

Established analogues of classical additive combinatorics results for group actions

Developed the concept of left invariant submodular functions in this context

Provided new proofs and reorganized the theoretical framework

Abstract

We establish analogues in the context of group actions or group representations of some classical problems and results in additive combinatorics of groups. We also study the notion of left invariant submodular function defined on power sets which plays a central role in our proofs.This new version is a completely reorganized version of the preceding one.