# An ergodic theorem for non-singular actions of the Heisenberg groups

## TL;DR

This paper establishes a non-singular ergodic theorem for the discrete Heisenberg group, identifying a specific sequence of subsets for which the theorem holds across all non-singular actions, using a special metric with the Besicovitch property.

## Contribution

It introduces a universal sequence of subsets for the Heisenberg group ensuring the non-singular ergodic theorem applies to any of its actions, utilizing a novel metric and proof techniques.

## Key findings

- Existence of a universal subset sequence for the Heisenberg group
- Application of a metric with the Besicovitch covering property
- Extension of Hochman's proof to this context

## Abstract

We show that there is a sequence of subsets of each discrete Heisenberg group for which the non-singular ergodic theorem holds. The sequence depends only on the group; it works for any of its non-singular actions. To do this we use a metric which was recently shown by Le Donne and Rigot to have the Besicovitch covering property and then apply an adaptation of Hochman's proof of the multiparameter non-singular ergodic theorem. An exposition of how one proves non-singular ergodic theorems of this type is also included, along with a new proof for one of the key steps.