Analysis highlights the significance of Khintchine's theorem in metric diophantine approximation, suggesting key breakthroughs, including the proof of the Duffin-Schaeffer conjecture.
In this note, we review the history of Khintchine's Theorem which is the foundation of metric Diophantine approximation, and discuss several generalizations and recent breakthroughs in this area. We focus particularly on the direction of the Duffin-Schaeffer Conjecture which was spectacularly proven in 2020. We present some simplified key ideas of the proof that can also be applied in various other areas of number theory.
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Manuel Hauke (2025) studied this question.
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