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Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture
Manuel Hauke-Treuer, James Maynard, Andrew Pollington
Source 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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