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:<math>\left| x- \frac{p}{q} \right| < \frac{\psi(q)}{|q|}.</math>
[[Aleksandr Khinchin]] proved in 1926 that if the series <math display="inline">\sum_{q} \psi(q) </math> diverges, then almost every real number (in the sense of [[Lebesgue measure]]) is <math>\psi</math>-approximable, and if the series converges, then almost every real number is not <math>\psi</math>-approximable. The circle of ideas surrounding this theorem and its relatives is known as ''metric Diophantine approximation'' or the ''metric theory of Diophantine approximation'' (not to be confused with height "metrics" in [[Diophantine geometry]]) or ''metric number theory''.
{{harvtxt|Duffin|Schaeffer|1941}} proved a generalization of Khinchin's result, and posed what is now known as the [[Duffin–Schaeffer conjecture]] on the analogue of Khinchin's dichotomy for general, not necessarily decreasing, sequences <math>\psi</math> . {{harvtxt|Beresnevich|Velani|2006}} proved that a [[Hausdorff measure]] analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker.
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