Metrical Diophantine approximation for quaternions

TL;DR

This paper extends classical Diophantine approximation theorems to quaternions, establishing measure-theoretic results analogous to Khintchine and Jarnik's theorems using lim sup set techniques.

Contribution

It introduces quaternionic analogues of key metrical Diophantine approximation theorems, expanding the scope of classical results to higher-dimensional algebraic structures.

Findings

Established quaternionic versions of Khintchine's theorem

Proved quaternionic Jarnik and Jarnik-Besicovitch theorems

Applied measure of lim sup sets to quaternions

Abstract

Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.