Title: 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 $W(f,\theta)$
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.