Metrical theorems for unconventional height functions

TL;DR

This paper extends classical metric theorems in Diophantine approximation to unconventional height functions, providing new analogues of Khintchine and Jarník-Besicovitch theorems and addressing open questions.

Contribution

It introduces and proves metric theorems for approximation using non-standard height functions, expanding the scope of classical Diophantine approximation results.

Findings

Established analogues of Khintchine's theorem for unconventional heights

Proved Jarník-Besicovitch type theorems in this new setting

Answered open questions posed by Fishman and Simmons (2017)

Abstract

In this paper, we consider the simultaneous approximation of real points by rational points with the error of approximation given by the functions of `non-standard' heights. We prove analogues of Khintchine and Jarn\'ik-Besicovitch theorems for this setting, thus answering some questions raised by Fishman and Simmons (2017).