How Many Intervals Cover A Point in Random Dyadic Covering?

A.H. Fan and J.P. Kahane

Abstract: We consider a random covering determined by a random variable $X$ of the space ${\Bbb D}= \{0,1\}^{\Bbb N}$. We are interested in the covering number $N_n(t)$ of a point $t \in {\Bbb D}$ by cylinders of lengths $\geq 2^{-n}$. It is proved that points in $\Bbb D$ are differently covered in the sense that the random sets $\{ t \in {\Bbb D}\dpt N_n(t) - b\,n \,\sim\, c\,n^\alpha\}$ are non-empty for a certain range of $b$, any real number $c$ and any $1/2<\alpha<1$. Actually, the Hausdorff dimensions of these sets are calculated. The method may be applied to the first percolation on an infinite and locally finite tree.

Keywords: Random covering; Hausdorff dimension; Indexed martingale; Peyrière measure.

Classification (MSC2000): 52A22, 52A45, 60D05.

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