Long-Standing Problem of 'Golden Ratio' and Other Irrational Numbers Solved with 'Magical...

On the Duffin-Schaeffer conjecture

Let $ψ:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from thepositive integers to the non-negative reals. Consider the set $\mathcal{A}$ ofreal numbers $α$ for which there are infinitely many reduced fractions$a/q$ such that $|α-a/q|\le ψ(q)/q$. If $\sum_{q=1}^\inftyψ(q)ϕ(q)/q=\infty$, we show that $\mathcal{A}$ has full Lebesgue measure.This answers a question of Duffin and Schaeffer. As a corollary, we alsoestablish a conjecture...