PORTUGALIAE MATHEMATICA Vol. 58, No. 1, pp. 5975 (2001) 

How Many Intervals Cover A Point in Random Dyadic Covering?A.H. Fan and J.P. KahaneDépartement de Mathématiques et Informatique, Université de Picardie Jules Verne33, Rue Saint Leu, 80039 Amiens Cedex 1  FRANCE Email: aihua.fan@upicardie.fr Analyse harmonique, Université de Paris Sud 91405 Orsay Cedex  FRANCE Email: jeanpierre.kahane@math.upsud.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 nonempty 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. Full text of the article:
Electronic version published on: 9 Feb 2006. This page was last modified: 27 Nov 2007.
© 2001 Sociedade Portuguesa de Matemática
