Direct and inverse results for popular differences in trees of positive dimension

TL;DR

This paper explores the structure of sets within trees of positive dimension by establishing analogues of density and popular differences, using ergodic theory and inverse theorems to connect combinatorial and dynamical properties.

Contribution

It formalizes a Furstenberg-Weiss correspondence principle for trees and develops new inverse theorems for return times in measure-preserving systems.

Findings

Established analogues for trees of results relating set density and structure.

Proved a new inverse theorem for non-ergodic systems.

Connected combinatorial data on trees to Markov process dynamics.

Abstract

We establish analogues for trees of results relating the density of a set $E \subset \mathbb{N}$, the density of its set of popular differences, and the structure of $E$. To obtain our results, we formalise a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting we use a recent result of the first author with Bj\"orklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.