Metric Diophantine Approximation: aspects of recent work

TL;DR

This paper reviews classical and recent advances in metric Diophantine approximation, including strengthened theorems and developments on approximation on manifolds, covering well, badly, and inhomogeneously approximable points.

Contribution

It summarizes recent improvements and new directions in metric Diophantine approximation, especially on manifolds, building on classical results.

Findings

Strengthening of classical Diophantine approximation theorems

Recent progress in approximation on manifolds

Analysis of well, badly, and inhomogeneously approximable points

Abstract

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.