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arXiv · 2604.22689

Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay

Seongmin Kim
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Abstract

Khintchine's theorem on the measure dichotomy for the set of $ψ$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ramírez conjectured that the monotonicity condition can be removed in the inhomogeneous $nm=2$ cases. In this paper, we resolve the $(n,m)=(1,2)$ case for $ψ$ satisfying a polynomial decay condition $ψ(q)=O(q^{-δ})$ for some $δ>0.$

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Seongmin Kim. 2026-04-24. Khintchine's theorem for inhomogeneous simultaneous approximation with polynomial decay. https://arxiv.org/abs/2604.22689

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