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Approximate subgroups with bounded VC-dimension

Gabriel Conant, Anand Pillay

Abstract

We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite subsets $A$ of arbitrary groups $G$ where $A$ has "small tripling" and bounded VC-dimension: Roughly speaking, up to a small error, $A$ will be a union of a bounded number of translates of a coset nilprogression of bounded rank and step (see Theorem 2.1). We also prove a stronger result in the setting of bounded exponent (see Theorem 2.2). Our results extend recent work of Martin-Pizarro, Palacín, and Wolf on finite stable sets of small tripling.

Approximate subgroups with bounded VC-dimension

Abstract

We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite subsets of arbitrary groups where has "small tripling" and bounded VC-dimension: Roughly speaking, up to a small error, will be a union of a bounded number of translates of a coset nilprogression of bounded rank and step (see Theorem 2.1). We also prove a stronger result in the setting of bounded exponent (see Theorem 2.2). Our results extend recent work of Martin-Pizarro, Palacín, and Wolf on finite stable sets of small tripling.

Paper Structure

This paper contains 31 sections, 45 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Small doubling in abelian groups
  3. Small tripling in arbitrary groups
  4. Bounded exponent
  5. Main results
  6. NIP and VC-dimension
  7. Sketch of the proofs
  8. The abelian case (and a note on bounds)
  9. Combinatorial preliminaries
  10. Notation and terminology
  11. Coset nilprogressions
  12. Set systems and VC-dimension
  13. NIP sets of small tripling are essentially approximate groups
  14. Model theoretic preliminaries
  15. Formulas and NIP
  16. ...and 16 more sections

Key Result

Theorem 1.1

Suppose $G$ is an abelian group and $A\subseteq G$ is a finite set with $k$-doubling. Then there is a proper coset progression $P\subseteq 2A-2A$ of rank $O_k(1)$ such that $A$ is covered by $O_k(1)$ translates of $P$.

Theorems & Definitions (130)

  • Theorem 1.1: Green & Ruzsa GrRuz
  • Theorem 1.2: Breuillard-Green-Tao BGT, Tao TaoPSE
  • Theorem 1.3
  • Theorem 2.1: main result
  • Theorem 2.2
  • Definition 3.1
  • Theorem 3.2: Tao TaoPSE
  • Remark 3.3
  • Definition 3.4: Generalized progression
  • Definition 3.5: Generalized arithmetic progression
  • ...and 120 more