Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture

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.