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PreprintCC BY-NC-SA 4.0arXiv

Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture

  • Manuel Hauke-Treuer
  • James Maynard
  • Andrew Pollington
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Abstract

The Duffin-Schaeffer Conjecture, proven in 2020, gives a zero-full dichotomy for the approximation of irrational numbers with reduced fractions for arbitrary approximation functions. We construct approximation functions such that the inhomogeneous analogue of the Duffin-Schaeffer Conjecture fails. The counterexamples are obtained for all non-zero rationals and certain Liouville numbers.

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Cite

BibTeXRIS
BibTeX
@misc{hauketreuer2026counterexamples,
  title = {{Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture}},
  author = {Manuel Hauke-Treuer and James Maynard and Andrew Pollington},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30921},
  eprint = {2609.30921},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30921},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture
AU  - Manuel Hauke-Treuer
AU  - James Maynard
AU  - Andrew Pollington
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30921
UR  - https://arxiv.org/abs/2609.30921
AB  - The Duffin-Schaeffer Conjecture, proven in 2020, gives a zero-full dichotomy for the approximation of irrational numbers with reduced fractions for arbitrary approximation functions. We construct approximation functions such that the inhomogeneous analogue of the Duffin-Schaeffer Conjecture fails. The counterexamples are obtained for all non-zero rationals and certain Liouville numbers.
ER  -