Diophantine approximation in metric space

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.