# A Fourier analytic approach to inhomogeneous Diophantine approximation

## TL;DR

This paper applies Fourier analysis to inhomogeneous Diophantine approximation, determining the Hausdorff dimension of certain approximation sets and providing conditions for their measure properties.

## Contribution

It introduces a Fourier analytic method to analyze inhomogeneous Diophantine approximation sets with rational numbers in reduced form, fully characterizing their Hausdorff dimension.

## Key findings

- Determines Hausdorff dimension of $W(f,\theta)$ in terms of $f$ and $\theta$.
- Provides a new sufficient condition for $W(f,\theta)$ to have full Lebesgue measure.
- Connects results to Diophantine approximation with additional constraints.

## Abstract

In this paper, we study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object to study is the set $W(f,\theta)$ as follows, \begin{eqnarray*} \left\{x\in [0,1]:\left |x-\frac{m+\theta(n)}{n}\right|<\frac{f(n)}{n}\text{ for infinitely many coprime pairs of numbers } m,n\right\}, \end{eqnarray*} where $\{f(n)\}_{n\in\mathbb{N}}$ and $\{\theta(n)\}_{n\in\mathbb{N}}$ are sequences of real numbers in $[0,1/2]$. We will completely determine the Hausdorff dimension of $W(f,\theta)$ in terms of $f$ and $\theta$. As a by-product, we also obtain a new sufficient condition for $W(f,\theta)$ to have full Lebesgue measure and this result is closely related to the study of \ds with extra conditions.