Inhomogeneous Duffin–Schaeffer behavior for arbitrary shifts

Characterize the behavior of the inhomogeneous Duffin–Schaeffer conjecture for arbitrary shifts, particularly for non-Liouville irrational shifts.

Background

The paper constructs counterexamples to the inhomogeneous Duffin–Schaeffer conjecture for every nonzero rational shift and for irrational shifts satisfying an exceptionally strong Diophantine approximation condition. The authors explicitly state that their method is restricted to exceptionally well-approximable shifts and identify the behavior for arbitrary, especially non-Liouville, shifts as unresolved.

References

However, it is clear that our approach is restricted to exceptionally well-approximable $\gamma$. The behaviour for arbitrary, in particular non-Liouville $\gamma$, remains widely open.

— Counterexamples to the inhomogeneous Duffin-Schaeffer Conjecture  (2609.30921 - Hauke-Treuer et al., 25 Sep 2026) in Section 1, Introduction and statement of results, Remark 1