Khinchin's Theorem and Extensions
Aleksandr Khinchin proved in 1926 that if is a non-increasing function from the positive integers to the positive real numbers such that then for almost all real numbers x (not necessarily algebraic), there are at most finitely many rational p/q and
Similarly, if the sum diverges, then for almost all real numbers, there are infinitely many such rational numbers p/q.
In 1941, R.J. Duffin and A.C. Schaeffer proved a more general theorem that implies Khinchin's result, and made a conjecture now known by their name as the Duffin–Schaeffer conjecture. In 2006, V. Beresnevich and S. Velani proved a Hausdorff measure analogue of the conjecture, published in the Annals of Mathematics.
Read more about this topic: Diophantine Approximation
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