Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture

Let $Q\subseteq \mathbb{N}$ be a subset, and let $\psi \text{\hspace{0.17em}⁣}:\mathbb{N}\to [0,\frac{1}{2})$, $\theta \text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{R}$ be functions. Let $\{A_q\}$ and $\{B_q\}$ be sequences of integers such that $\gcd (A_q,B_q)=1$ and $B_q>0$ for all $q$. Define $W_Q^{\ast }(\psi ,\theta )$ to be the set of $x\in [0,1]$ for which $$\mid x-\frac{p+\theta (q)}{q}\mid <\frac{\psi (q)}{q}$$ holds for infinitely many $(p,q)\in \mathbb{Z}\times Q$ with $\gcd (B_{q}p+A_q,q)=1$. In this paper, we determine the Fourier dimension of $W_Q^{\ast }(\psi ,\theta )$. Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case $\psi (q)=q^{-\tau }$ with $\tau \ge 1$) and the one-dimensional version of a result by Cai and Hambrook on the inhomogeneous approximable set, but also provides a complete inhomogeneous generalization. Moreover, it gives an affirmative answer to the coprime formulation of the Chen–Xiong conjecture.