Structure and regularity for subsets of groups with finite VC-dimension

C Terry,G Conant,Anand Pillay

doi:10.4171/jems/1111

Abstract

Suppose $G$ is a finite group and $A\\subseteq G$ is such that ${gA:g\\in G}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\\epsilon>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $\\epsilon$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao \[8] and Hrushovski \[28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon \[1] on approximate homomorphisms.

Highlights

Introduction and statement of resultsSzemeredi’s Regularity Lemma [47] is a fundamental result about graphs, which has found broad applications in graph theory, computer science, and arithmetic combinatorics
We prove the theorems about finite groups by taking ultraproducts of counterexamples in order to obtain infinite pseudofinite groups contradicting the companion theorems
We include the details for the sake of clarity and to observe that the method works for Bohr neighborhoods in nonabelian finite groups defined using compact metric groups

Summary

Introduction and statement of results

Szemeredi’s Regularity Lemma [47] is a fundamental result about graphs, which has found broad applications in graph theory, computer science, and arithmetic combinatorics. In [13], the first two authors proved “generic compact domination” for the quotient group G/G0θ0r , where θr(x; y, u) := θ(x · u; y) and G0θ0r is the intersection of all θr-type-definable bounded-index subgroups of G (see Definition 2.12) In this case, G/G0θ0r is a compact Hausdorff group, and generic compact domination roughly states that if A ⊆ G is θr-definable, the set of cosets of G0θ0r , which intersect both A and G\A in “large” sets with respect to the pseudofinite counting measure, has Haar measure 0 (see Theorem 2.14). This follows our theme, as such groups satisfy a strong form of compact domination (see Lemma 6.5).

Preliminaries

Structure and regularity: the profinite case

Bohr neighborhoods

Structure and regularity: the general case

Distal arithmetic regularity

Compact p-adic analytic groups