Intersection spaces and multiple transverse recurrence

TL;DR

This paper investigates multiple recurrence phenomena in measure-preserving group actions on Polish spaces, establishing a new recurrence theorem under conditions related to Delone sets and uniform approximate lattices.

Contribution

It introduces a multiple transverse recurrence theorem for unimodular lcsc group actions, linking recurrence properties to the structure of return time sets as Delone sets.

Findings

Established a multiple transverse recurrence theorem.

Connected recurrence properties to Delone set structures.

Applied results to uniform approximate lattices.

Abstract

We study multiple recurrence properties along separated cross sections for pmp actions of unimodular lcsc group on Polish spaces. We establish a multiple transverse recurrence theorem under the assumption that sufficiently large powers of the return time set are Delone sets. Typical examples of such situations arise from the theory of uniform approximate lattices.