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Claim. J.-P. Kahane, Sur le recouvrement d'un cercle par des arcs disposés au hasard, C. R. Acad. Sci. Paris 248 (1959), 184–186, is a note of the séance of 5 January 1959. It places arcs of given lengths $1>l_1\ge l_2\ge\cdots$ independently and uniformly on the circle of unit length. Its Theorem 1 proves that the arcs cover the whole circle almost surely as soon as

$$\limsup_{t\to\infty}\frac{1}{\log t}\sum_{l_n>1/t}l_n>1.$$

Theorem 2 shows that the constant $1$ cannot be lowered: for every $\varepsilon>0$, some sequence whose lim sup exceeds $1-\varepsilon$ does not cover almost surely. Theorem 3 proves that covering almost surely requires $\limsup_{n\to\infty}l_n\,n\log\log n>0$. The note says that its results do not decide $l_n=\alpha/n$ with $\alpha\le1$. The site credits Kahane with $a_n=(1+c)/n$, $c>0$. As written, $a_1=1+c$ exceeds the circumference, so the first arc alone covers the circle. The case has content for lengths equal to $(1+c)/n$ from some index on. There the sum in Theorem 1 is $(1+c)\log t+O(1)$, so the lim sup is $1+c>1$ (a remark of this page). The paper has no library card.

Covers. The nonincreasing sequences of lengths below $1$ that satisfy Theorem 1's condition, which cover almost surely; they include lengths $(1+c)/n$ from some index on. Also the sequences with $l_n\,n\log\log n\to0$, which do not cover almost surely (Theorem 3). Not covered: $l_n=\alpha/n$ with $\alpha\le1$, which the note leaves open and the site credits to Erdős (unpublished), and the criterion itself, which is Shepp's.

Depends on. No page of this wiki.

Acceptance. Refereed: the note appeared in C. R. Acad. Sci. Paris 248 (1959). The site's curator, T. F. Bloom, labels the problem SOLVED and credits Kahane with the case $a_n=(1+c)/n$ (problem page accessed); the label settles the problem through Shepp's criterion, not this note, so the page lists no reviewed evidence.