On the Duffin-Schaeffer conjecture

Let ψ : N → R>0 be an arbitrary function from the positive integers to the nonnegative reals. Consider the set A of real numbers α for which there are infinitely many reduced
fractions a/q such that |α − a/q| 6 ψ(q)/q. If P∞
q=1 ψ(q)ϕ(q)/q = ∞, we show that A has
full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also
establish a conjecture due to Catlin regarding non-reduced solutions to the inequality |α − a/q| 6
ψ(q)/q, giving a refinement of Khinchin’s Theorem.