Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture

Let $Q \subseteq \mathbb{N}$ be a subset, and let $\psi\colon \mathbb{N} \to [0, \tfrac{1}{2})$, $\theta\colon \mathbb{N} \to \mathbb{R}$ 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 $$\left| x - \frac{p + \theta(q)}{q} \right| < \frac{\psi(q)}{q}$$ 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.