Diophantine approximation in metric space

TL;DR

This paper extends Diophantine approximation to arbitrary totally bounded metric spaces by introducing a hierarchy of 'abstract rationals' and establishing dimension bounds, broadening classical number theory concepts.

Contribution

It proposes a novel model for Diophantine approximation in metric spaces using abstract rationals, and proves new dimension bounds within this framework.

Findings

Established Jarnik-Besicovitch type dimension bounds

Demonstrated the sharpness of these bounds

Extended classical approximation concepts to metric spaces

Abstract

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread' points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.