An heterogeneous ubiquity theorem, application to self-similar measures with overlaps

TL;DR

This paper develops a general mass transference principle for measures, including self-similar measures with overlaps, extending the understanding of measure distribution from balls to arbitrary open sets.

Contribution

It introduces a broad mass transference principle applicable to self-similar measures without separation conditions, broadening previous results.

Findings

Established a mass transference principle for self-similar measures with overlaps.

Extended the principle to arbitrary open sets under general conditions.

Provided new tools for analyzing measure distribution in fractal geometry.

Abstract

In this article, one investigates in a very general frame mass transference principles from ball to arbitrary open sets when the sequence of balls is distributed according to a finite measure. As an application of the main theorem, a mass transference principle is established when the measure is self-similar with no separation conditions.