A mass transference principle and sets with large intersections

TL;DR

This paper establishes a mass transference principle for general shapes, demonstrating that certain limsup-sets belong to classes of large intersection sets, extending previous results to more general geometric configurations.

Contribution

It introduces a new mass transference principle applicable to general shapes and proves the limsup-sets are in classes of large intersection sets, a novel extension.

Findings

Limsup-sets belong to classes of large intersection sets.

The proof uses Vitali's covering lemma and Riesz energies.

Extends previous results to more general shapes.

Abstract

I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer. This has previously not been proved for as general shapes as in this paper.