Inhomogeneous Diophantine approximation with general error functions

TL;DR

This paper investigates the Hausdorff dimension of sets defined by inhomogeneous Diophantine approximation with general error functions, providing sharp estimates based on the Diophantine type of the irrational number.

Contribution

It offers new sharp estimations for the Hausdorff dimension of inhomogeneous approximation sets considering general decreasing error functions and the Diophantine type.

Findings

Derived precise Hausdorff dimension bounds for approximation sets.

Extended results to a broad class of error functions.

Connected Diophantine type with approximation set complexity.

Abstract

Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R: |n\al -y| < \varphi(n) \text{for infinitely many} n},] where $|\cdot|$ denotes the distance to the nearest integer.