A Fourier analytic approach to inhomogeneous Diophantine approximation

Abstract

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

changes been made according to various suggestions