Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finding product sets in some classes of amenable groups

Dimitrios Charamaras, Andreas Mountakis

TL;DR

The paper tackles the problem of finding structured sumsets inside large subsets of amenable groups by introducing square absolute continuity (SAC) and exploiting Følner-density notions. It proves that SAC groups with positive density yield an infinite subset B of A such that B \triangleleft B is contained in a shifted version of A, generalizing Erdős-type results beyond the integers. The authors extend SAC to key classes (e.g., torsion-free finitely generated nilpotent groups, virtually nilpotent groups, and abelian groups with finite-index doubling), and derive strong corollaries for Mal'cev coordinates and doubling subgroups, including an optimality discussion via counterexamples in H_3. The approach blends dynamical correspondence principles, a Kronecker-factor analysis on homogeneous spaces, and a continuous ergodic decomposition to convert combinatorial density statements into dynamical configurations, broadening the scope of product-set phenomena in amenable groups with potential implications for additive combinatorics in non-abelian settings.

Abstract

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erdős about sumsets in large subsets of the natural numbers. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups, and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group $G$ from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in\mathbb{N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in a recent survey of Kra, Moreira, Richter and Robertson.

Finding product sets in some classes of amenable groups

TL;DR

The paper tackles the problem of finding structured sumsets inside large subsets of amenable groups by introducing square absolute continuity (SAC) and exploiting Følner-density notions. It proves that SAC groups with positive density yield an infinite subset B of A such that B \triangleleft B is contained in a shifted version of A, generalizing Erdős-type results beyond the integers. The authors extend SAC to key classes (e.g., torsion-free finitely generated nilpotent groups, virtually nilpotent groups, and abelian groups with finite-index doubling), and derive strong corollaries for Mal'cev coordinates and doubling subgroups, including an optimality discussion via counterexamples in H_3. The approach blends dynamical correspondence principles, a Kronecker-factor analysis on homogeneous spaces, and a continuous ergodic decomposition to convert combinatorial density statements into dynamical configurations, broadening the scope of product-set phenomena in amenable groups with potential implications for additive combinatorics in non-abelian settings.

Abstract

In 2022, using methods from ergodic theory, Kra, Moreira, Richter, and Robertson resolved a longstanding conjecture of Erdős about sumsets in large subsets of the natural numbers. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups, and all abelian groups with the property that the subgroup has finite index. We prove that in any group from the above classes, any with positive upper Banach density contains a shifted product set of the form , for some infinite sequence and some . In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in a recent survey of Kra, Moreira, Richter and Robertson.
Paper Structure (16 sections, 41 theorems, 188 equations)

This paper contains 16 sections, 41 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. More product sets and open questions
  4. Proof ideas
  5. Preliminaries
  6. Reduction of Theorem \ref{['mt1']} to dynamical statements
  7. Dynamical reformulation via correspondence principle
  8. Erdős progressions
  9. Continuous factor maps to the Kronecker factor
  10. Measures on Erdős progressions and the proof of \ref{['dr3']}
  11. Measures on Erdős progressions
  12. A continuous ergodic decomposition
  13. The proof of \ref{['dr3']}
  14. Proof of the corollaries of \ref{['mt1']}
  15. Counterexamples on product sets
  16. ...and 1 more sections

Key Result

Theorem 1.6

Let $G$ be a square absolutely continuous group and $A\subset G$ with positive left upper Banach density. Then there exist an infinite sequence $B=(b_n)_{n\in \mathbb{N}}\subset A$ and some $t\in G$ such that

Theorems & Definitions (89)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 79 more