Problem #526: Let $a_n\geq 0$ with $a_n\to 0$ and $\sum a_n=\infty$

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Progress

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0$ has this property, which Erdős (unpublished) improved to $a_n=\frac{1}{n}$.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\[\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.\]