Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 58, No. 1, pp. 59-75 (2001)

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How Many Intervals Cover A Point in Random Dyadic Covering?

A.H. Fan and J.P. Kahane

Département de Mathématiques et Informatique, Université de Picardie Jules Verne
33, Rue Saint Leu, 80039 Amiens Cedex 1 -- FRANCE
E-mail: ai-hua.fan@u-picardie.fr
Analyse harmonique, Université de Paris Sud
91405 Orsay Cedex -- FRANCE
E-mail: jean-pierre.kahane@math.u-psud.fr

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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Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.

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