Inhomogeneous Khintchine–Groshev theorem without monotonicity

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-06-04 DOI:10.1112/blms.70114

Seongmin Kim

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Abstract

The Khintchine–Groshev theorem in Diophantine approximation theory says that there is a dichotomy of the Lebesgue measure of sets of $ψ$ -approximable numbers, given a monotonic function $ψ$ . Allen and Ramírez removed the monotonicity condition from the inhomogeneous Khintchine–Groshev theorem for cases with $n m ⩾ 3$ and conjectured that it also holds for $n m = 2$ . In this paper, we prove this conjecture in the case of $(n, m) = (2, 1)$ . We also prove it for the case of $(n, m) = (1, 2)$ with a rational inhomogeneous parameter.

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无单调性非齐次Khintchine-Groshev定理

丢芬图近似理论中的Khintchine-Groshev定理说，给定一个单调函数ψ $\psi$, ψ $\psi$ -可近似数集合的勒贝格测度有一个二分法。Allen和Ramírez在n m大于或等于3 $nm\geqslant 3$的情况下从非齐次Khintchine-Groshev定理中去除单调性条件，并推测它也适用于n m = 2$nm=2$。本文在(n, m) = (2,1) $(n,m)=(2,1)$的情况下证明了这个猜想。对于(n, m) = (1,2) $(n,m)=(1,2)$具有有理非齐次参数的情况，我们也证明了这一点。

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