A note on the Duffin-Schaeffer conjecture

الأصل · EN

Given a sequence of real numbers {ψ(n)}n with 0≤ ψ(n)<1, let W(ψ) denote the set of x∈[0,1] for which |xn-m|<ψ(n) for infinitely many coprime pairs (n,m). The purpose of this note is to show that if there exists an ε>0 such that ∑nψ(n)¹⁺ε·φ(n)/n=∞, then the Lebesgue measure of W(ψ) equals 1.