Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

TL;DR

This paper develops a comprehensive measure and Hausdorff dimension theory for inhomogeneous Diophantine approximation on curves in Euclidean space, extending classical homogeneous results to inhomogeneous settings.

Contribution

It introduces a unified framework for inhomogeneous approximation on curves, generalizing key measure-theoretic and Hausdorff dimension results from homogeneous theory.

Findings

Generalized measure-theoretic theorems for inhomogeneous approximation

Established Hausdorff dimension results for planar curves

Extended classical homogeneous theorems to inhomogeneous cases

Abstract

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in $R^n$ akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978), Dodson, Dickinson (2000) and Beresnevich, Bernik, Kleinbock, Margulis (2002). In the case of planar curves, the complete Hausdorff dimension theory is developed