论文标题
托里矩阵转换的目标问题收缩：重新访问标准问题
The Shrinking Target Problem for Matrix Transformations of Tori: revisiting the standard problem
论文作者
论文摘要
令$ t $为具有真实系数的$ d \ times d $矩阵。然后，$ t $确定$ d $ -dimensional torus $ {\ bbb t}^d = {\ mathbb {r}}^d/{\ bbb z}^d $的自图。令$ \ {e_n \} _ {n \ in \ mathbb {n}} $是$ {\ bbb t}^d $的一系列序列，让$ w（t，\ {e__n \ \}）$是点$ \ \ \ \ \ \ \ \ \ \ \ n x} \ bb bb in $ t^n（\ mathbf {x}）\在{\ mathbb {n}} $中无限的$ n \ in e_n $中。对于一大批子集（即，那些满足所谓有限属性$（{\ boldsymbol {\ rm b}}}）$，其中包括球，矩形和高粘性物），我们表明$ d $ d $ d $ dobesional-demensional lebesgue的lebesgie lebesgue衡量了一个缩小目标$ w（t，t，t，t，e y zer $ zer zer zer zer zer）。 （分解）。实际上，我们证明了该零一个标准的定量形式，该标准描述了计数函数的渐近行为$ r（x，n）：= \＃\ big \ {1 \ le n \ le n \ le n \ le n：t^{n}（n}（x）\ in E_n \} $。计数结果利用了一个通用的定量语句，该语句适用于大型类测量的动力系统（即，那些满足所谓的汇总混合属性的属性）。接下来，我们将注意力转向$ w（t，\ {e_n \}）$的Hausdorff尺寸。在这种情况下，$ e_n $的子集是球，矩形或倍曲底，我们获得了尺寸的精确公式。这些形状分别对应于经典二磷抗菌近似的同时，加权和乘法理论。球的尺寸结果概括了希尔在较早的一篇论文中获得的尺寸和第三名以整数矩阵的作者的真实矩阵。在最后一部分中，我们讨论了本文中证明的结果的各种问题。
Let $T$ be a $d\times d$ matrix with real coefficients. Then $T$ determines a self-map of the $d$-dimensional torus ${\Bbb T}^d={\mathbb{R}}^d/{\Bbb Z}^d$. Let $ \{E_n \}_{n \in \mathbb{N}} $ be a sequence of subsets of ${\Bbb T}^d$ and let $W(T,\{E_n \})$ be the set of points $\mathbf{x} \in {\Bbb T}^d$ such that $T^n(\mathbf{x})\in E_n $ for infinitely many $n\in {\mathbb{N}}$. For a large class of subsets (namely, those satisfying the so called bounded property $ ({\boldsymbol{\rm B}}) $ which includes balls, rectangles, and hyperboloids) we show that the $d$-dimensional Lebesgue measure of the shrinking target set $W(T,\{E_n \})$ is zero (resp. one) if a natural volume sum converges (resp. diverges). In fact, we prove a quantitative form of this zero-one criteria that describes the asymptotic behaviour of the counting function $R(x,N):= \# \big\{ 1\le n \le N : T^{n}(x) \in E_n \} $. The counting result makes use of a general quantitative statement that holds for a large class measure-preserving dynamical systems (namely, those satisfying the so called summable-mixing property). We next turn our attention to the Hausdorff dimension of $W(T,\{E_n \})$. In the case the subsets $E_n$ are balls, rectangles or hyperboloids we obtain precise formulae for the dimension. These shapes correspond, respectively, to the simultaneous, weighted and multiplicative theories of classical Diophantine approximation. The dimension results for balls generalises those obtained in an earlier paper by Hill and the third-named author for integer matrices to real matrices. In the final section, we discuss various problems that stem from the results proved in the paper.
