普适系统与度量数论
Ubiquitous systems and metric number theory
摘要：
我们研究了$$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{μ,α}\times\Z^d\},$$的大小和大相交性质，其中$d\in\N$，$t\geq 1$，$I$是可数集，$(x_{i},r_{i})_{i\in I}$是$[0,1]^d\times (0,\infty)$中的一个族，$I^{μ,α}$表示所有$i\in I$的集合，使得对于给定的Borel测度$μ$和给定的$α>0$，具有中心$x_{i}$和半径$r_{i}$的球的$μ$-质量表现为${r_{i}}^α$.我们建立了集合$E_{t}$属于关于某个规范函数$h$的大交集合类$\grint^h(\R^d)$，只要$(x_{i},r_{i})_{i\in I}$是关于$μ$的异构泛在系统.特别地，$E_{t}$对于在零的邻域中比$h$增加得更快的每个规范函数$g$具有无限Hausdorff $g$-测度。我们也给出了度量数论的几个应用。
英文摘要：
We investigate the size and large intersection properties of $$E_{t}=\{x\in\R^d \:|\: \|x-k-x_{i}\|<{r_{i}}^t\text{for infinitely many}(i,k)\in I^{μ,α}\times\Z^d\},$$ where $d\in\N$, $t\geq 1$, $I$ is a denumerable set, $(x_{i},r_{i})_{i\in I}$ is a family in $[0,1]^d\times (0,\infty)$ and $I^{μ,α}$ denotes the set of all $i\in I$ such that the $μ$-mass of the ball with center $x_{i}$ and radius $r_{i}$ behaves as ${r_{i}}^α$ for a given Borel measure $μ$ and a given $α>0$. We establish that the set $E_{t}$ belongs to the class $\grint^h(\R^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_{i},r_{i})_{i\in I}$ is a heterogeneous ubiquitous system with respect to $μ$. In particular, $E_{t}$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory.
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