Classical metric Diophantine approximation revisited

TL;DR

This paper reviews recent progress and open problems in classical metric Diophantine approximation, focusing on measure theoretic approaches, and discusses generalizations of key conjectures like Duffin-Schaeffer and Catlin.

Contribution

It presents new formulations and explorations of generalizations of longstanding conjectures in metric Diophantine approximation.

Findings

Discussion of recent progress in the field

Formulation of generalized conjectures

Identification of open problems

Abstract

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored.