Inhomogeneous Fiber Refinement of Gallagher's Theorem

Abstract

Let \(s>0\), \(α,γ\in\mathbb{R}\), and let \(ψ\colon \mathbb{N}\to\mathbb{R}_{>0}\) be a non-increasing function. Consider the limsup set \[ W_s(ψ,α,γ) = \left\{ β\in[0,1] \colon \norm{nβ-γ} < \frac{ψ(n)}{\norm{nα}^{\,s}} \ \text{ for infinitely many } n\in\mathbb{N} \right\}. \] Chow [Duke J. Math. 2018] settled the measure-theoretic result for \(s=1\). In this paper, we investigate the Lebesgue measure of \(W_s(ψ,α,γ)\) for general \(s>0\).