[#P2936] Complete coverage by random disks at the harmonic scale

Problem. Let \(D=\{x\in\mathbb R^2:\|x\|\le1\}\), let \(X_1,X_2,\ldots\) be independent uniform points of \(D\), and let \(D_n\) be the closed disk centered at \(X_n\) with radius \(n^{-1/2}\). Is \(\Pr(D\subseteq\bigcup_{n\ge1}D_n)=1\)?

An illustration accompanying the statement of Complete coverage by random disks at the harmonic scale. — An illustration accompanying the statement. The statement alone defines the problem.

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Definitions and notation

1Context

Multiscale nets, empty-cell bounds, and boundary-strip estimates can be reused at later scales. The harmonic sum makes fixed-point coverage easy while leaving the geometry of exceptional holes unresolved.

2Problem setup

Definition 1. Uniform on D means normalized planar Lebesgue measure, and the centers X_n are mutually independent.

Remark 1. Complete coverage requires every point of the uncountable closed disk D to lie in at least one random disk D_n.

3What counts as a solution

Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability.
A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature. Exact unresolved remainder: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.^[2][3][1]

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Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature. Exact unresolved remainder: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature. Exact unresolved remainder: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.^[2][3][1] Relevance to this problem For Complete coverage by random disks at the harmonic scale, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 3, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature. Exact unresolved remainder: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail. Open full record →
Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail.^[1] Relevance to this problem For Complete coverage by random disks at the harmonic scale, record harmonic-random-disk-cover-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove that the union covers D almost surely, or prove that the event of an uncovered point has positive probability. A computer-assisted argument must supply certified finite-net bounds and a rigorous passage that controls holes between net points and the boundary for the infinite tail. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-27. The comments establish almost-sure coverage of each fixed point and zero area of the uncovered set. They do not establish complete coverage of the uncountable disk, and the exact critical sequence was not settled in the checked random-covering literature.

Penrose's arXiv:2101.06306 and related finite random geometric coverage results were checked. They use finite samples and threshold radii with boundary corrections, which do not directly decide this infinite nonidentical-radius process.
Dvoretzky covering references were checked for the one-dimensional analogue. The planar boundary and two-dimensional holes require different capacity or net estimates.
A local corpus search for harmonic random disks, n^{-1/2} covering, and complete random coverage found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. For any fixed x in the interior of D, the sum of Pr(x in D_n) diverges at harmonic order, so x is covered almost surely. This pointwise statement permits a random exceptional point.

2See also

Monotonicity of consecutive-adjacency avoidance in random permutationsprobability
Eventual monotonicity in a signed subset-sum local limitprobability
Eventual decrease for distinct cycle lengths in random permutationsprobability

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“Complete coverage by random disks at the harmonic scale.” TheoremDB. P2936. Problem statement; statement text SHA-256 3c2cc7c59da20e71fe7fa3ccfcf89e2c99822a47ddf897ce3ff7b9e40ae14bfc. https://theoremdb.org/statement/?ref=P2936

BibTeX

@misc{theoremdb-problem-3c2cc7c59da20e71fe7fa3ccfcf89e2c99822a47ddf897ce3ff7b9e40ae14bfc,
  title = {{Complete coverage by random disks at the harmonic scale}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 3c2cc7c59da20e71fe7fa3ccfcf89e2c99822a47ddf897ce3ff7b9e40ae14bfc},
  url = {https://theoremdb.org/statement/?ref=P2936}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. MathOverflow question 458793, “Complete coverage by random disks at the harmonic scale,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 458793 and every visible answer and comment were checked on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Complete coverage by random disks at the harmonic scale; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Complete coverage by random disks at the harmonic scale, the reviewed source scope is Full question, answers, and visible comments concerning Complete coverage by random disks at the harmonic scale; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
arXiv preprint 2101.06306, linked primary source for “Complete coverage by random disks at the harmonic scale,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2101.06306, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and limit theorems for the coverage threshold R_n of n uniform centers with common radius.Source used to assess the problem's recorded status.For Complete coverage by random disks at the harmonic scale, this source gives finite-sample Euclidean coverage asymptotics and boundary effects; it does not settle complete coverage by radii 1/sqrt(n).
J.-P. Kahane, “Dvoretzky problem,” Encyclopedia of Mathematics, revision 51207, last edited January 3, 2021. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗website · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at definition and criterion for Dvoretzky's random covering problem on the circle.Source used to assess the problem's recorded status.For Complete coverage by random disks at the harmonic scale, this source supplies general random-covering context in a different geometry and with different covering sets.

An original CC0 reformulation motivated by the cited MathOverflow random-covering question; the radius and coverage event are stated directly.

[#P2936] Complete coverage by random disks at the harmonic scale

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