论文标题
关于不均匀二芬太汀近似的公制理论：ERDőS-VAALER型结果
On the metric theory of inhomogeneous Diophantine approximation: An Erdős-Vaaler type result
论文作者
论文摘要
1958年，Szüsz被证明是Khintchine在Diophantine近似方面的不均匀版本。 szüsz的定理指出，对于任何非进攻近似函数$ψ：\ mathbb {n} \ to（0,1/2）$带有$ \ sum_qψ（q）= \ infty $和任何数字$γ和任何数字$γ，$ \ [ w（ψ，γ）= \ {x \ in [0,1]：| Qx-p-γ| <ψ（q） \] 具有完整的Lebesgue度量。从那时起，放松单调性条件的结果很少。在本文中，我们表明，如果$γ$不能很好地通过合理数量近似，那么单调性条件可以被上限条件$ψ（q）= o（q（q（q（\ log log q）^2）^2）^{ - 1} $取代。 \ sqrt {2}。$通常，如果$γ$是不合理的，则$ψ（q）= o（q^{ - 1}（\ log \ log \ log \ log q）^{ - 2}）$，此外， \ [ \ left（\ liminf_ {q \ to \ infty} \ sum_ {q = q}^{q^{（\ log q） \] 然后$ w（ψ，γ）$具有完整的lebesgue度量。我们的证明是基于对非理性旋转差异的定量研究。
In 1958, Szüsz proved an inhomogeneous version of Khintchine's theorem on Diophantine approximation. Szüsz's theorem states that for any non-increasing approximation function $ψ:\mathbb{N}\to (0,1/2)$ with $\sum_q ψ(q)=\infty$ and any number $γ,$ the following set \[ W(ψ,γ)=\{x\in [0,1]: |qx-p-γ|< ψ(q) \text{ for infinitely many } q,p\in\mathbb{N}\} \] has full Lebesgue measure. Since then, there are very few results in relaxing the monotonicity condition. In this paper, we show that if $γ$ is can not be approximate by rational numbers too well, then the monotonicity condition can be replaced by the upper bound condition $ψ(q)=O((q(\log\log q)^2)^{-1}).$ In particular, this covers the case when $γ$ is not Liouville, for example $π,e,\ln 2, \sqrt{2}.$ In general, if $γ$ is irrational, $ψ(q)=O(q^{-1}(\log\log q)^{-2})$ and in addition, \[ \left(\liminf_{Q\to\infty} \sum_{q=Q}^{Q^{(\log Q)^{1/8} }}ψ(q)\right)=\infty, \] then $W(ψ,γ)$ has full Lebesgue measure. Our proof is based on a quantitative study of the discrepancy for irrational rotations.
