VC-dimension of generalized progressions in some nonabelian groups

TL;DR

This paper investigates the VC-dimension of generalized progressions in certain nonabelian groups, providing bounds that enhance understanding of their complexity in additive combinatorics.

Contribution

It introduces bounds on the VC-dimension of generalized progressions in specific nonabelian groups, linking combinatorial structure with learning theory concepts.

Findings

Finite upper bounds on VC-dimension in free groups

Finite upper bounds on VC-dimension in the Heisenberg group

Implications for the study of approximate groups

Abstract

We analyze generalized progressions in some nonabelian groups using a measure of complexity called VC-dimension, which was originally introduced in statistical learning theory by Vapnik and Chervonenkis. Here by a "generalized progression" in a group $G$, we mean a finite subset of $G$ built from a fixed set of generators in analogy to a (multidimensional) arithmetic progression of integers. These sets play an important role in additive combinatorics and, in particular, the study of approximate groups. Our two main results establish finite upper bounds on the VC-dimension of certain set systems of generalized progressions in finitely generated free groups and also the Heisenberg group over $\mathbb{Z}$.