Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 625-633

A NOTE ON THE HITTING PROBABILITIES OF RANDOM COVERING SETS

Bing Li and Ville Suomala

South China University of Technology, Department of Mathematics
Guangzhou, 510641, P.R. China; and
University of Oulu, Department of Mathematical Sciences
P.O. Box 3000, 90014 University of Oulu, Finland; libing0826 'at' gmail.com

University of Oulu, Department of Mathematical Sciences
P.O. Box 3000, 90014 University of Oulu, Finland; ville.suomala 'at' oulu.fi

Abstract. Let E = lim sup_{n
\to\infty}(g_n + \xi_n) be the random covering set on the torus T^d, where {g_n} is a sequence of ball-like sets and \xi_n is a sequence of independent random variables uniformly distributed on T^d. We prove that E \cap F \neq \emptyset almost surely whenever F\subset T^d is an analytic set with Hausdorff dimension, dim_H(F) > d - \alpha, where \alpha is the almost sure Hausdorff dimension of E. Moreover, examples are given to show that the condition on dim_H(F) cannot be replaced by the packing dimension of F.

2010 Mathematics Subject Classification: Primary 60D05, 28A78, 28A80.

Key words: Random covering sets, hitting probability, Hausdorff dimension.

Reference to this article: B. Li and V. Suomala: A note on the hitting probabilities of random covering sets. Ann. Acad. Sci. Fenn. Math. 39 (2014), 625-633.

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doi:10.5186/aasfm.2014.3927