论文标题
Hausdorff量度的Dirichlet非可不良仿射形式的度量
Hausdorff measure of sets of Dirichlet non-improvable affine forms
论文作者
论文摘要
对于降低实际有价值的函数$ψ$，A对$（a，\ mathbf {b}）$ a $ $ m \ times n $ matrix $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ $$ \ | a \ MATHBF {Q}+\ MATHBF {B} - \ MATHBF {p} \ |^m <m <ψ（t）\ quad \ text {and} \ quad \ | \ | \ | \ mathbf {q} $ \ mathbf {q} \ in \ mathbb {z}^n $对于所有足够大的$ t $，其中$ \ | \ cdot \ | $表示至上的标准。 Kleinbock and Wadleigh（2019）为$ψ$ -Dirichlet nonnon-dromprovable集的Lebesgue度量建立了一个集成性标准。在本文中，我们证明了与$ψ$ -Dirichlet不可挽回集合的Hausdorff度量的类似标准。另外，我们将此结果扩展到单一的度量案例，即$ \ mathbf {b} $已修复。作为一个应用程序，我们计算了一组对$（a，\ mathbf {b}）$的Hausdorff尺寸，并带有均匀的Diophantine指数$ \ wideHat {w}（a，\ mathbf {b}）\ leq w $。
For a decreasing real valued function $ψ$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $ψ$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m < ψ(T)\quad\text{and}\quad\|\mathbf{q}\|^n < T$$ has a solution $\mathbf{p}\in\mathbb{Z}^m$, $\mathbf{q}\in\mathbb{Z}^n$ for all sufficiently large $T$, where $\|\cdot\|$ denotes the supremum norm. Kleinbock and Wadleigh (2019) established an integrability criterion for the Lebesgue measure of the $ψ$-Dirichlet non-improvable set. In this paper, we prove a similar criterion for the Hausdorff measure of the $ψ$-Dirichlet non-improvable set. Also, we extend this result to the singly metric case that $\mathbf{b}$ is fixed. As an application, we compute the Hausdorff dimension of the set of pairs $(A,\mathbf{b})$ with uniform Diophantine exponents $\widehat{w}(A,\mathbf{b})\leq w$.
