Problem 526

Let $a_n\geq 0$ with $a_n\to 0$ and $\sum a_n=\infty$ . Find a necessary and sufficient condition on the $a_n$ such that, if we choose (independently and uniformly) random arcs on the unit circle of length $a_n$ , then all the circle is covered with probability $1$ .

Source: erdosproblems.com/526

Probability
Geometry

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved. The site labels the problem SOLVED and credits Shepp's necessary and sufficient condition [Sh72], recorded as the accepted claim Shepp 1972. Kahane's sufficient and necessary conditions [Ka59], which give the case $a_n=(1+c)/n$ the site credits to him, are the accepted partial claim Kahane 1959; Erdős's cases $a_n=1/n$ and $a_n=(1-c)/n$ are unpublished and have no claim page.

Problem 526

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