Fourier Dimension in Inhomogeneous Duffin--Schaeffer Conjecture

Abstract

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.