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Equidistribution in 2-Nilpotent Polish Groups and triple restricted sumsets

Ethan Ackelsberg, Asgar Jamneshan

Abstract

The aim of this paper is to establish a Ratner-type equidistribution theorem for orbits on homogeneous spaces associated with $2$-nilpotent locally compact Polish groups under the action of a countable discrete abelian group. We apply this result to establish the existence of triple restricted sumsets in subsets of positive density in arbitrary countable discrete abelian groups, subject to a necessary finiteness condition.

Equidistribution in 2-Nilpotent Polish Groups and triple restricted sumsets

Abstract

The aim of this paper is to establish a Ratner-type equidistribution theorem for orbits on homogeneous spaces associated with -nilpotent locally compact Polish groups under the action of a countable discrete abelian group. We apply this result to establish the existence of triple restricted sumsets in subsets of positive density in arbitrary countable discrete abelian groups, subject to a necessary finiteness condition.

Paper Structure

This paper contains 16 sections, 46 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Equidistribution in 2-nilpotent Polish groups
  3. Triple restricted sumsets in abelian groups
  4. Proof of the equidistribution result
  5. Nilpotent translational systems are distal
  6. Minimal components of translational systems are uniquely ergodic
  7. Ergodic decomposition of translational systems
  8. Proof of Theorem \ref{['thm: main']}
  9. Proof of the sumset result
  10. A dynamical encoding of sumsets
  11. Useful facts from ergodic theory
  12. Progressive measures
  13. Erdős progressions in nilpotent translational systems
  14. Lifting to a progressive measure
  15. Completing the proof
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $\Gamma$ be a countable discrete abelian group, and suppose $(X, \Sigma_X, \mu_X, T_X)$ is an ergodic $\Gamma$-system. The following are equivalent:

Theorems & Definitions (99)

  • Definition 1.1
  • Theorem 1.2: jst
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Definition 1.5
  • Theorem 1.6: kmrr_finite_sums
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • ...and 89 more