Covering the circle with random open sets (Q1766449)

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 \textit{A. Dvoretzky} [Proc.\ Nat.\ Acad.\ Sci.\ USA 42, 199--203 (1956; Zbl 0074.12301)], was solved completely by \textit{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.