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Covering the circle with random open sets. (English) Zbl 1073.60009
Place a sequence of arcs of decreasing length \(\{\ell_n\}_{n\geq1}\), \(0<\ell_n<1\), uniformly and independently on a circumference of unit length. Under what condition on the \(\ell_n\)’s will the circle be covered by the union of these arcs almost surely? The problem, raised by A. Dvoretzky [Proc. Nat. Acad. Sci. USA 42, 199–203 (1956; Zbl 0074.12301)], was solved completely by L. A. Shepp [Isr. J. Math. 11, 328-345 (1972; Zbl 0241.60008)], who showed that a necessary and sufficient condition for the circle to be covered almost surely is the divergence of the series with terms \(e^{\ell_1+\cdots+\ell_n}/n^2\). This paper addresses an extension of the Dvoretzky problem in which random arcs are replaced by random translates of open sets, each containing a finite number of arcs separated by a positive distance. A necessary and sufficient condition is given.
MSC:
60D05 Geometric probability and stochastic geometry
60F15 Strong limit theorems
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