Hotspot lemmas for non-compact spaces

TL;DR

This paper generalizes the hotspot lemma to dynamical systems on non-compact spaces, addressing previous proof issues and providing a complete criterion for normality in this broader context.

Contribution

It extends the hotspot lemma to non-compact spaces and corrects prior proof issues, adding a necessary and sufficient condition for the criterion.

Findings

Extended the hotspot lemma to non-compact spaces

Identified and fixed issues in previous proofs

Provided a complete necessary and sufficient condition

Abstract

We extend the hotspot lemma, or Piatetski\u{i}-Shapiro normality criterion, to dynamical systems on non-compact spaces. We build on the work of N. G. Moshchevitin and I. D. Shkredov, noting an issue in some of their proofs, and adding a necessary and sufficient condition to remove this issue.