# P2936: Complete coverage by random disks at the harmonic scale

- ID: `P2936`
- Reference: `harmonic-random-disk-cover`
- Page: https://theoremdb.org/statements/P2936
- Record maturity: Reviewed problem with recorded work

## 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 \(\Pr(D\subseteq\bigcup_{n\ge1}D_n)=1\)?

### Problem setup

- **Definition.** Uniform on D means normalized planar Lebesgue measure, and the centers X_n are mutually independent.
- **Remark.** Complete coverage requires every point of the uncountable closed disk D to lie in at least one random disk D_n.

### What 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.

## Status

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](#reference-2) [3](#reference-3) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- 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.
- The MathOverflow question and every visible answer and comment were checked on 2026-07-27. Borel-Cantelli handles each fixed point, and Fubini gives zero uncovered area, while neither argument controls all points simultaneously.
- 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.

- Recorded example: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `harmonic-random-disk-cover`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 458793, “Complete coverage by random disks at the harmonic scale,” checked 2026-08-01. Question 458793 and every visible answer and comment were checked on 2026-07-27. https://mathoverflow.net/questions/458793/will-a-unit-disk-be-completely-covered-by-randomly-placed-disks-of-area-pi-fr
   - 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.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Mathew D. Penrose, “Random Euclidean coverage from within,” Probability Theory and Related Fields 185(3-4) (2023), 747-814. DOI 10.1007/s00440-022-01182-5. abstract and limit theorems for the coverage threshold R_n of n uniform centers with common radius https://arxiv.org/abs/2101.06306
   - preprint; reference source; arXiv:2101.06306, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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).
3. <a id="reference-3"></a>J.-P. Kahane, “Dvoretzky problem,” Encyclopedia of Mathematics, revision 51207, last edited January 3, 2021. definition and criterion for Dvoretzky's random covering problem on the circle https://encyclopediaofmath.org/wiki/Dvoretzky_problem
   - website; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.