Duffin–Schaeffer conjecture

The Duffin–Schaeffer conjecture is an important conjecture in metric number theory proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{+}}$ is a real-valued function taking on positive values, then for almost all ${\displaystyle \alpha }$ (with respect to Lebesgue measure), the inequality

has infinitely many solutions in co-prime integers ${\displaystyle p,q}$ with ${\displaystyle q>0}$ if and only if the sum

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