Dimensions of random covering sets in Riemann manifolds
Dimensions of random covering sets in Riemann manifolds
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DOI：
10.1214/17-aop1210
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发表时间：
2015-08
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通讯作者：
De-Jun Feng;E. Jarvenpaa;M. Jarvenpaa;Ville Suomala
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文献类型：
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作者：
De-Jun Feng;E. Jarvenpaa;M. Jarvenpaa;Ville Suomala
设${\pmb M}$${\pmb N}$和${\pmb K}$是$d$维黎曼流形。设${\bf A}:=(A_n)_{n\in{\Bbb N}}$是满足必要密度条件的${\pmb M}$的Lebesgue可测子集序列，${\bf x}:=(x_n)_{n\in {\Bbb N}}$是根据相对于黎曼体积的非纯奇异测度分布在${\pmb K}$上的独立随机变量序列。我们给出了随机覆盖集的Hausdorff维数的几乎确定值${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$的一个公式。这里$A_n(x_n)$是一个依赖于$x_n$的$A_n$的微分同构图像。我们还验证了${\bf E}({\bf x},{\bf A})$的包装尺寸几乎肯定等于$d$。
Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ according to a measure which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$. Here $A_n(x_n)$ is a diffeomorphic image of $A_n$ depending on $x_n$. We also verify that the packing dimensions of ${\bf E}({\bf x},{\bf A})$ equal $d$ almost surely.