Problem 526

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claims/: The 2 claim pages of Problem 526, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $1$.

Formulation. The standing answers the site's wording, read as its sources read it. The "unit circle" is the circle of circumference one, the setting of Dvoretzky [Dv56], Kahane [Ka59] and Shepp [Sh72]. On a circle of radius one every length would be divided by $2\pi$, and the site's covering example $a_n=1/n$ would no longer cover. The arcs are placed independently with uniform positions, so whether they cover depends only on the multiset of lengths, not on the order in which the $a_n$ are listed. A sequence with a length of at least $1$ is covered with probability one. The accepted claim states how Shepp's criterion answers the question under this reading.

Status. 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the site credits to him, are the accepted partial claim Kahane 1959; Erdős's cases $a_n=1/n$ and $a_n=(1-c)/n$ are unpublished and have no claim page.

Source. erdosproblems.com/526, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #526, https://www.erdosproblems.com/526.

References.

• [Dv56] Dvoretzky, Aryeh, On covering a circle by randomly placed arcs. Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 199-203.
• [Ka59] Kahane, Jean-Pierre, Sur le recouvrement d'un cercle par des arcs disposés au hasard. C. R. Acad. Sci. Paris 248 (1959), 184-186.
• [Sh72] Shepp, L. A., Covering the circle with random arcs. Israel J. Math. 11 (1972), no. 3, 328-345.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• dvoretzky_1956_covering_circle_randomly_placed_arcs
• dvoretzky_1956_covering_circle_randomly_placed_arcs / theorem_1
• dvoretzky_1956_covering_circle_randomly_placed_arcs / theorem_2