Variants of Khintchine's theorem in metric Diophantine approximation

Abstract

New results towards the Duffin-Schaeffer conjecture, which is a fundamental problem in metric number theory, have been established recently assuming extra divergence. Given a non-negative function ψ:N→R we denote by W(ψ) the set of all x∈R such that |nx−a|<ψ(n) for infinitely many a,n. Analogously, denote W′(ψ) if we additionally require a,n to be coprime. Aistleitner et al. [2] proved that W′(ψ) is of full Lebesgue measure if there exist an ε>0 such that ∑n=2∞ψ(n)φ(n)/(n(log⁡n)ε)=∞. This result seems to be the best one can expect from the method used. Assuming the extra divergence ∑n=2∞ψ(n)/(log⁡n)ε=∞ we prove that W(ψ) is of full measure. This could also be deduced from the result in [2], but we believe that our proof is of independent interest, since its method is totally different from the one in [2]. As a further application of our method, we prove that a variant of Khintchine's theorem is true without monotonicity, subject to an additional condition on the set of divisors of the support of ψ.