# A measure theoretic result for approximation by Delone sets

## TL;DR

This paper explores measure theoretic approximation properties of Delone sets, establishing a converse of the Borel-Cantelli lemma, and extends classical Diophantine approximation concepts to these sets.

## Contribution

It introduces a measure theoretic framework for Delone sets and proves a full converse of the Borel-Cantelli lemma in this context.

## Key findings

- Established a measure theoretic approximation property for Delone sets
- Proved a full converse of the Borel-Cantelli lemma for these sets
- Extended classical Diophantine approximation results to Delone sets

## Abstract

With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish a full converse of the Borel-Cantelli lemma. This provides an analogue of more classical problems in the metric theory of Diophantine approximation, but with the distance to the nearest integer function replaced by distance to an arbitrary Delone set.