Inhomogeneous Fiber Refinement of Gallagher's Theorem

Qian Li; Bo Tan; Qing-Long Zhou

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arXiv · 2611.01781

Inhomogeneous Fiber Refinement of Gallagher's Theorem

Qian Li·Bo Tan·Qing-Long Zhou

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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 $s>0$.

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Qian Li, Bo Tan, Qing-Long Zhou. 2026-11-01. Inhomogeneous Fiber Refinement of Gallagher's Theorem. https://arxiv.org/abs/2611.01781

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