The Duffin-Schaeffer conjecture with extra divergence

Christoph Aistleitner*, Thomas Lachmann, Marc Alexandre Munsch, Niclas Technau, Agamemnon Zafeiropoulos

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function ψ : N → R for almost all reals x there are infinitely many coprime solutions (a, n) to the inequality |nx − a| < ψ(n), provided that the series ∑∞ n=1 ψ(n)φ(n)/n is divergent. In the present paper we prove that the conjecture is true under the “extra divergence” assumption that divergence of the series still holds when ψ(n) is replaced by ψ(n)/(log n)ε
for some ε > 0. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.
Original languageEnglish
Article number106808
Number of pages11
JournalAdvances in Mathematics
Publication statusPublished - 2019


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