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Abstract

We study inhomogeneous Diophantine approximation with rational numbers of reduced form. The central object is the set $$ W(f,\theta)= \bigg\{x\in [0,1]:\bigg |x-\frac{m+\theta(n)}{n}\bigg| \lt \frac{f(n)}{n}\quad\ \text{for infinitely many coprime pairs } m,n\bigg\}, $$ 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; this result is closely related to the Duffin–Schaeffer conjecture with extra conditions.