Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts
Year: 2026
Type: preprint
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 $\mathbb R$.
Source: arXiv (Cornell University)
Authors: He, Yubin, Liao, Lingmin
Institutions: Shantou University, Wuhan University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G5982710811
OpenAlex ID: W7214606883