VC-dimension of generalized progressions in some nonabelian groups
Gabriel Conant, Aycin Iplikci Arodirik, Tora Ozawa, David Zeng
TL;DR
This paper studies the VC-dimension of set systems of generalized progressions in nonabelian groups, connecting combinatorial geometry with group theory and model theory. It provides finite upper bounds for two nonabelian settings: the Heisenberg group $\mathbb{H}$ (explicit bound $\le 267$ for the progression system and $\le 140$ for individual nilprogressions) and finitely generated free groups $F_k$ (bound $\le 3k-1$). The analysis combines model-theoretic NIP/semialgebraic methods for $\mathbb{H}$ with geometric/group-theoretic tree/pseudometric techniques for $F_k$, illustrating how generalized progressions can be controlled in nonabelian environments and informing the broader program of structure theorems for approximate groups. The results contribute to understanding how combinatorial complexity, via VC-dimension, interacts with nilprogressions and Bohr-type structures in nonabelian groups, with implications for arithmetic regularity and approx-subgroup theory.
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 $\mathbb{Z}$.