Almost periodic discrete sets

TL;DR

This paper provides a geometric framework for understanding almost periodic sets in Euclidean space using a special metric, establishing their completeness and linking them to almost periodic measures.

Contribution

It introduces a new geometric description of almost periodic sets, proves the space's completeness, and relates these sets to almost periodic measures.

Findings

Complete space of almost periodic sets established

Analogue of Bochner criterion proved

Connection between almost periodic sets and measures shown

Abstract

Using a special metric in the space of sequences, we give a geometric description of almost periodic sets in the $k$-dimensional Euclidean space. We prove the completeness of the space of almost periodic sets and some analogue of the Bochner criterion of almost periodicity. Also, we show the connection between these sets and almost periodic measures.