An anisotropic inhomogeneous ubiquity Theorem

TL;DR

This paper extends mass transference principles in metric number theory to cases with anisotropic, inhomogeneous measures and rectangular approximating sets, broadening the scope of previous results.

Contribution

It introduces a new ubiquity theorem for rectangles and quasi-Bernoulli measures, combining shape flexibility with measure inhomogeneity.

Findings

Established a ubiquity theorem for rectangular approximations

Demonstrated applicability to quasi-Bernoulli measures

Extended mass transference principles to anisotropic inhomogeneous settings

Abstract

Recently, mass transference principles in metric number theory extend towards two direction. On one hand, the shape of the approximating sets can be taken of various shape, balls, rectangles or even general open sets (one refers to some results of Rams and Koivusalo regarding this last example) when the ambient measure is Lebesgue, on the other hand, progress have been made to understand what can be said when the ambient measure is changed. For instance some computation in the case of a self-similar measure with open set condition have been made by Barral and Seuret. This article mixes the two approaches and presents a ubiquity theorem which handles the case where the approximating sets are rectangles and the measure is quasi-Bernoulli, but fully supported.