论文标题
关于乘二芬太丁近似的度量理论
On the metric theory of multiplicative Diophantine approximation
论文作者
论文摘要
1962年，加拉格尔（Gallagher）证明了Khintchine定理关于二磷酸近似的较高维度。 Gallagher的定理指出，对于任何非增加近似函数$ψ：\ mathbb {n} \ to（0,1/2）$带有$ \ sum_ {q = 1}^{\ infty}^{\ infty}ψ（q） \ [ \ {（x，y）\ in [0,1]^2：\ | qx-γ\ | \ | | qy-γ'\ | <ψ（q） \] 具有完整的Lebesgue度量。最近，Chow和Technau证明了上述结果的完全不均匀版本（对$γ，γ'$无限制）。 在本文中，我们证明了纤维乘二磷酸近似的ERDőS-VAALER型结果。在此过程中，通过一种不同的方法，我们证明了Chow-Technau的定理的版本稍弱，条件是至少有$γ，γ'$不是Liouville。我们还扩展了Chow-technau的结果，以实现纤维不均匀的Gallagher的Liouville纤维定理。
In 1962, Gallagher proved an higher dimensional version of Khintchine's theorem on Diophantine approximation. Gallagher's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_{q=1}^{\infty} ψ(q)\log q=\infty$ and $γ=γ'=0$ the following set \[ \{(x,y)\in [0,1]^2: \|qx-γ\|\|qy-γ'\|<ψ(q) \text{ infinitely often}\} \] has full Lebesgue measure. Recently, Chow and Technau proved a fully inhomogeneous version (without restrictions on $γ,γ'$) of the above result. In this paper, we prove an Erdős-Vaaler type result for fibred multiplicative Diophantine approximation. Along the way, via a different method, we prove a slightly weaker version of Chow-Technau's theorem with the condition that at least one of $γ,γ'$ is not Liouville. We also extend Chow-Technau's result for fibred inhomogeneous Gallagher's theorem for Liouville fibres.
