A Note on Random Coverings of Tori

TL;DR

This paper generalizes previous results on the dimension of limsup-sets in random coverings of tori, establishing a larger intersection property under weaker assumptions, thus strengthening the theoretical understanding of such sets.

Contribution

It introduces a broader, more powerful result on the dimension and intersection properties of limsup-sets in random coverings of tori, extending prior work with weaker conditions.

Findings

Provides a larger intersection property for limsup-sets

Weakens assumptions needed for dimension results

Includes previous results as a special case

Abstract

This note provides a generalisation of a recent result by J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala, (to appear), on the dimension of limsup-sets of random coverings of tori. The result in this note is stronger in the sense that it provides also a large intersection property of the limsup-sets, the assumptions are weaker, and it implies the result of J\"arvenp\"a\"a, J\"arvenp\"a\"a, Koivusalo, Li, and Suomala as a special case. The proof is based on a recent result by Persson and Reeve from 2013.