Problem 999

For any function $f:\mathbb{N}\to \mathbb{N}$ the property that, for almost all $\alpha$

\left\lvert \alpha-\frac{p}{q}\right\rvert < \frac{f(q)}{q}

has infinitely many solutions with $(p,q)=1$ , is equivalent to

\sum_{q\geq 1}\phi(q)\frac{f(q)}{q}=\infty.

Source: erdosproblems.com/999

Number theory
Diophantine approximation

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SolvedProved

An accepted solution exists. The statement is true.

Proved. The site's label is PROVED, crediting Koukoulopoulos and Maynard's 2019 proof of the Duffin-Schaeffer conjecture, refereed in 2020 and recorded on its claim page (Koukoulopoulos and Maynard, 2019).

Problem 999

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Proof claims

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