Sets with large intersection properties in metric spaces

TL;DR

This paper extends the concept of al-sets, characterized by large intersection properties and Hausdorff dimension, from Euclidean spaces to more general metric spaces, aiding in Diophantine approximation analysis.

Contribution

It generalizes the characterization of al-sets with large intersection properties to broader metric spaces beyond Euclidean settings.

Findings

al-sets are characterized in general metric spaces.

These sets have Hausdorff dimension at least s.

They are closed under countable intersections.

Abstract

In this work we reproduce the characterization of $\Gg^s$-sets from the euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric spaces. These sets have Hausdorff dimension at least $s$ and are closed by countable intersections, which is particularly useful to estimate the dimension of the so called sets of $\alpha$-approximable points (that typically appear in Diophantine approximations).