Inhomogeneous Diophantine approximation on planar curves

TL;DR

This paper advances the metric theory of inhomogeneous Diophantine approximation on planar curves, extending homogeneous results and focusing on the distribution of shifted rational points near these curves.

Contribution

It develops the inhomogeneous metric theory for points on planar curves, building upon and incorporating recent homogeneous Khintchine-Jarnik theorems.

Findings

Established optimal results on the distribution of shifted rational points near planar curves

Extended homogeneous Diophantine approximation theorems to the inhomogeneous setting

Unified inhomogeneous and homogeneous approximation theories in a planar context

Abstract

The inhomogeneous metric theory for the set of simultaneously $\psi$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann. of Math. (2), 166 (2007), pp. 367-426] and [Invent. Math., 166 (2006), pp. 103-124]. The key lies in obtaining essentially the best possible results regarding the distribution of `shifted' rational points near planar curves.