An Inhomogeneous Jarn\'ik type theorem for planar curves

TL;DR

This paper proves an inhomogeneous Jarník type theorem for planar curves, completing the metric theory of Diophantine approximation on these curves in both homogeneous and inhomogeneous cases.

Contribution

It establishes the missing inhomogeneous Jarník theorem for planar curves, advancing the understanding of metric Diophantine approximation on manifolds.

Findings

Proves inhomogeneous Jarník theorem for planar curves

Completes the metric theory for approximation on planar curves

Bridges the gap in inhomogeneous Diophantine approximation results

Abstract

In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarn\'ik are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarn\'ik type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarn\'ik type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves.