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arXiv · 2609.30870

Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts

Yubin He·Lingmin Liao
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Abstract

Let $θ\in\mathbb Q\setminus\{0\}$. We construct an approximating function $ψ:\mathbb N\to[0,\frac12)$ for which \[ \sum_{q=1}^{\infty}\frac{φ(q)}{q}ψ(q)=\infty, \] but the set of $x\in[0,1]$ for which \[ |qx-a-θ|<ψ(q),\qquad \gcd(a,q)=1, \] holds for infinitely many $(a,q)\in\mathbb Z\times\mathbb N$ has Lebesgue measure zero. Thus the inhomogeneous analogue of the Duffin--Schaeffer conjecture fails for every nonzero rational shift. By a modification of the construction, we further show that the set of shifts for which the inhomogeneous Duffin--Schaeffer conjecture fails is residual in $\mathbb R$.

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Yubin He, Lingmin Liao. 2026-09-25. Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts. https://arxiv.org/abs/2609.30870

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