SOLVED
This has been resolved in some other way than a proof or disproof.
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 $1$.
A problem of Dvoretzky
[Dv56]. It is easy to see that (under the given conditions alone) almost all the circle is covered with probability $1$.
Kahane
[Ka59] showed that $a_n=\frac{1+c}{n}$ with $c>0$ has this property, which Erdős (unpublished) improved to $a_n=\frac{1}{n}$. Erdős also showed that $a_n=\frac{1-c}{n}$ with $c>0$ does not have this property.
Solved by Shepp
[Sh72], who showed that a necessary and sufficient condition is that\[\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.\]
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #526, https://www.erdosproblems.com/526, accessed 2026-01-16
