论文标题
在较高维度的迪里奇莱特非不良矩阵集的一般豪斯多夫度量
Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions
论文作者
论文摘要
令$ψ：\ mathbb r _ {+} \ to \ mathbb r _ {+} $为非信息功能。一对$（a，\ mathbf b），$ a $是一个真正的$ m \ times n $矩阵和$ \ mathbf b \ in \ mathbb r^{m}，如果系统$ \ | a \ a \ a \ m m马理\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \ | \ Mathbf Q \ |^n <t $$可在$ \ Mathbf p \ in \ Mathbb z^{m}中解决，$ $ \ Mathbf q \ in \ Mathbb z^{n} $用于所有足够大的$ t $ t $ where $ t $ where $ \ cdot \ cdot \ cdot \ cdot \ | $ deNotes the supremummum norm。对于$ψ$ -Dirichlet不可销售的套件，Kleinbock - Wadleigh（2019）证明了Lebesgue测量标准，而Kim-Kim（2021）建立了Hausdorff量度的结果。在本文中，我们获得了Kim-kim（2021）$ψ$ -Dirichlet不可销售的Kim-Kim（2021）结果的广义Hausdorff $ f $ -Measure版本。
Let $ψ:\mathbb R_{+}\to \mathbb R_{+}$ be a nonincreasing function. A pair $(A,\mathbf b),$ where $A$ is a real $m\times n$ matrix and $\mathbf b\in\mathbb R^{m},$ is said to be $ψ$-Dirichlet improvable, if the system $$\|A\mathbf q +\mathbf b-\mathbf p\|^m<ψ(T), \quad \|\mathbf q\|^n<T$$ is solvable in $\mathbf p\in\mathbb Z^{m},$ $\mathbf q\in\mathbb Z^{n}$ for all sufficiently large $T$ where $\|\cdot\|$ denotes the supremum norm. For $ψ$-Dirichlet non-improvable sets, Kleinbock--Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim--Kim (2021) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff $f$-measure version of Kim--Kim (2021) results for $ψ$-Dirichlet non-improvable sets.
