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Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts
Abstract
Let θ∈ℚ∖{0}. We construct an approximating function ψ:ℕ→[0,1/2) for which ∑q=1∞φ(q)/qψ(q)=∞, but the set of x∈[0,1] for which |qx−a−θ|<ψ(q), gcd(a,q)=1, holds for infinitely many (a,q)∈ℤ×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 ℝ.
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BibTeX
@misc{he2026counterexamples,
title = {{Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts}},
author = {Yubin He and Lingmin Liao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.30870},
eprint = {2609.30870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30870},
}
RIS
TY - UNPB
TI - Counterexamples to the inhomogeneous Duffin-Schaeffer conjecture for a residual set of shifts
AU - Yubin He
AU - Lingmin Liao
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2609.30870
UR - https://arxiv.org/abs/2609.30870
AB - Let θ∈ℚ∖{0}. We construct an approximating function ψ:ℕ→[0,1/2) for which ∑_q=1^∞φ(q)/qψ(q)=∞, but the set of x∈[0,1] for which |qx−a−θ|<ψ(q), gcd(a,q)=1, holds for infinitely many (a,q)∈ℤ×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 ℝ.
ER -