Dimension of random limsup sets.(English)Zbl 1429.28012
Let \(\omega_0, \omega_1, \ldots\) be independent uniform random points in \([0,1]^d\), for a positive integer \(d\). For a sequence \(r_0, r_1, \ldots\)of positive real numbers, consider the random set\[ E = \bigcap_{m \geq 0} \bigcup_{n \geq m} B (\omega_n, r_n ) ,\]where \(B(x,r)\) is the Euclidean \(d\)-ball with centre \(x\) and radius \(r\). Set \(t_0 = \inf \{ t \geq 0 : \sum_{n \geq 0} r_n^t < \infty \}\).A known theorem states that the Hausdorff dimension of \(E\) is equal to \(\min ( t_0, d)\), almost surely,a result obtained for \(d=1\) byA.-H. Fan andJ. Wu [Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 1, 125–131 (2004;Zbl 1037.60010)] and which can be derived for general \(d\) by an argument ofA. Durand et al. [“On randomly placed arcs on the circle”, in: Recent developments in fractals and related fields. Boston, MA: Birkhäuser. 343–351 (2010)].In the paper under review, the author gives a new proof of this theorem.
Reviewer: Andrew Wade (Durham)
Citations:
Zbl 1037.60010Cite
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References:
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