The Duffin-Schaeffer conjecture with extra divergence

Abstract

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.MSC classification: 11J83, 11K55, 11K60.

Khintchine's Theorem With Random Fractions

“…In the most recent of these, Aistleitner et al . show that if for some the sum diverges, then has full measure . Of course, the Duffin–Schaeffer conjecture predicts that this statement is still true if the ‐factor is removed.…”

“…diverges, then W ( ) has full measure [2]. Of course, the Duffin-Schaeffer conjecture predicts that this statement is still true if the (log n) ε -factor is removed.…”

“…Remark. Theorem 1.8 (and the two results that follow it) may be considered in the context of recent works [1,2,4,13], which establish the Duffin-Scheffer conjecture under "extra divergence" assumptions. In the most recent of these, Aistleitner et al show that if for some ε > 0 the sum…”

“…It is not known whether Catlin's conjecture implies the Duffin-Schaeffer conjecture. 2 (It was claimed so in [6], but an error was found by Vaaler [18], and the issue remains unresolved.) Regardless, one might expect that even if Catlin's conjecture is weaker than the Duffin-Schaeffer conjecture, it may still be very difficult to prove, and therefore Conjecture 1.6 might also be comparably difficult.…”

A Fourier-analytic approach to inhomogeneous Diophantine approximation

“…Duffin-Schaeffer conjecture with extra conditions: known results before [1]. A lot of work has been done since the birth of the above conjecture.…”