Dvoretzky 1956 covering circle randomly placed arcs

../

theorem_1: Dvoretzky's sufficient condition for random arcs to cover every point of a circle of unit circumference infinitely often with probability one, for a nonincreasing sequence of lengths; lengths at least 2 log i / i for all large i satisfy it.

theorem_2: Dvoretzky's theorem that divergence of the sum of the arc lengths does not make random arcs cover the whole circle with probability one, proved by a nonincreasing sequence of lengths constant on rapidly growing blocks.

---

Aryeh Dvoretzky, On covering a circle by randomly placed arcs, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 199--203. Communicated by P. A. Smith, January 19, 1956; Hebrew University of Jerusalem and Columbia University.

The copy read for this card is a journal scan of the five printed pages with a text layer (physical PDF p. $n$ is printed p. $198+n$); p. 199 opens with the end of the preceding article and p. 203 carries the note's footnotes 2--5 and the start of the following article, and the formulas in the text layer are garbled, so the statements below were checked on the page images of pp. 200--201. Provenance: downloaded in September 2026; the download URL was not recorded; 432,923 bytes. No notice is printed, the Crossref record names no license, and the publisher's article page and rights page could not be read on 2026-10-02 (DOI 10.1073/pnas.42.4.199, HTTP 403); the term is unstated.

Read status. Claims checked: Section I and Theorems 1 and 2 were read clause by clause (the prose in the text layer, the formulas on the page images); the two short proofs were read but not verified.

Contents

The paper poses the problem page's question and proves the two results below.

• Setting (Section I, pp. 199--200): $C$ is a circle of unit circumference, $a_i$ ($i\ge1$) are positive numbers less than $1$, and $A_i$ are arcs of length $a_i$ whose centers are independent and uniform on $C$. By the Borel--Cantelli lemma, a fixed point lies in infinitely many $A_i$ with probability $1$ if and only if $\sum a_i=\infty$ (2); by Fubini's theorem, (2) is necessary and sufficient for the arcs to cover almost all of $C$ with probability $1$. The paper asks whether the "almost all" can be dropped, answers no, and states that "we do not know necessary and sufficient conditions on the rate of divergence of $\sum a_i$ which would insure that all of $C$ is covered with probability 1" (p. 200).
• Theorem 1 (p. 200; proof pp. 200--201 by dividing $C$ into cells and the occupancy formula (10)): for a nonincreasing sequence $(a_i)$ with $\varlimsup_{i\to\infty}\bigl(ia_i-2\log(1/a_i)\bigr)>-\infty$ (5), almost surely every point of $C$ lies in infinitely many of the arcs, $P\{\text{all of }C\text{ covered i.o.}\}=1$ (6); the paper notes that $a_i\ge2i^{-1}\log i$ from some index on suffices.
• Theorem 2 (p. 201; proof pp. 201--202 by a monotone sequence built from a rapidly increasing sequence of integers, (20)--(21)): for some sequences $(a_i)$ with divergent sum (2), the arcs fail with positive probability to cover all of $C$, $P\{C\subseteq\bigcup_iA_i\}<1$ (15); the paper states this equivalently as $P\{\text{all of }C\text{ covered i.o.}\}=0$ (16). The proof "can easily be modified to yield an explicit slow divergence rate", which the paper does not do.
• Section IV (p. 202): remarks on improvements, on nonuniformly distributed centers, and on an application to the everywhere divergence on $|z|=1$ of almost all random sign series $\sum\pm b_iz^i$.

Compiled scope

The whole five-page note was read, with the statements of Theorems 1 and 2 checked on the page images; the proofs were followed but not verified. Nothing here is independently reviewed.

Bears on. #526: the paper states (p. 200) that it does not know a necessary and sufficient condition on the rate of divergence of $\sum a_i$ for the whole circle to be covered with probability $1$, the condition the problem asks for. Theorem 2 gives lengths that are positive, tend to $0$ and have divergent sum, yet leave part of the circle uncovered with positive probability, so the problem's hypotheses alone do not ensure covering; Theorem 1 gives a sufficient condition for nonincreasing lengths. Neither theorem answers the problem; the criterion the problem page records is Shepp's later one (the page's [Sh72]).

Results. Theorem 1 (p. 200), the sufficient condition (5) for nonincreasing lengths; Theorem 2 (p. 201), lengths with divergent sum that fail to cover the whole circle with positive probability.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.