# P2546: A centered maximal-function halo constant for three intervals

- ID: `P2546`
- Reference: `centered-maximal-three-interval-halo`
- Page: https://theoremdb.org/statements/P2546
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/centered-maximal-three-interval-halo?representation=markdown
- Record maturity: Prospecting problem with recorded work

## The problem

For a measurable set \(A\) that is a union of at most three bounded intervals in \(\mathbb R\) and satisfies \(0<|A|<\infty\), let \(M_c1_A(x)=\sup_{r>0}(2r)^{-1}\int_{x-r}^{x+r}1_A(y)\,dy\). Determine \(H_3=\sup_A |\{x:M_c1_A(x)\ge1/3\}|/|A|\).

### Context

The problem asks for the exact worst halo expansion at level \(1/3\) among sets with at most three interval components.

### Conventions

- **Convention.** Intervals may touch, and zero-length components may be discarded.
- **Convention.** Vertical bars denote Lebesgue measure.

### What counts as a solution

- Determine \(H_3\) exactly by giving an exact value or characterization, an extremizing or asymptotically extremizing interval configuration, and a proof of the matching global bound. Numerical upper and lower bounds alone are partial computational evidence.

## Status

**Resolution status:** Open.

**Packet-reported status.**

The packet certifies \(61/20\le H_3\le(11+\sqrt{61})/4\); the exact value of \(H_3\) remains open. [1](#reference-1)

**Recorded scope.** the centered Hardy-Littlewood halo at threshold 1/3 for one explicit union of three bounded intervals

[Read this exact record and its supporting evidence](/records/cm3ih-claim-rational-lower-bound)


*This research-packet status has not passed editorial review.*

## Research packet

### Evidence for the packet-reported status

**Computation 1 (An exact three-interval construction gives H_3 at least 61/20).** The packet certifies \(61/20\le H_3\le(11+\sqrt{61})/4\); the exact value of \(H_3\) remains open.

Put
\[
A=[0,1/5]\cup[3/4,21/20]\cup[11/5,27/10].
\]
Its component lengths are \(1/5,3/10,1/2\), so \(|A|=1\). For a consecutive block of components from \([a_i,b_i]\) through \([a_j,b_j]\), let \(m_{ij}\) be its total occupied length. Every point in
\[
K_{ij}=[b_j-3m_{ij}/2,\ a_i+3m_{ij}/2]
\]
has a centered interval containing that block whose average of \(\mathbf 1_A\) is at least \(1/3\). The six block intervals are
\[
[-1/10,3/10],\ [3/10,3/4],\ [3/5,6/5],\ [6/5,3/2],\ [3/2,39/20],\ [39/20,59/20].
\]
Their union is \([-1/10,59/20]\), of length \(61/20\). Hence
\[
H_3\geq 61/20=3.05.
\]
This is a certified lower bound. It does not establish that the displayed interval is the entire halo.

### Background and intake notes

- Original intake status: Status not established. No literature search was performed. Sharp one-dimensional maximal inequalities are well studied and may imply this restricted value.
- The endpoint geometry reduces any fixed interval configuration to finitely many candidate averaging radii, suggesting a semialgebraic optimization.
- The attractive route sums halo lengths component by component. The obstruction is that a centered averaging interval can bridge two or three components and create halo that belongs to no single component estimate.

- Recorded example: For one interval, the level-1/3 halo has twice the interval's measure, so \(H_3\ge2\).

### Other known results

- **Proposition 1** (supported): The sharp weak (1,1) constant for the centered maximal operator yields the rigorous upper bound 4.702562418976664. [1](#reference-1)

### Open directions

- **Route 1** (conjectured): Current certified bounds are 61/20 <= H_3 <= (11+sqrt(61))/4; proving an exact optimum requires a finite global analysis of the possible halo configurations. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (executable material): A standard-library Python script computes all six block halos, merges them, and checks the ratio with Fraction arithmetic. [1](#reference-1)

### Computational notes

- The one-interval lower bound follows exactly by scaling \(A\) to \([-1,1]\): the level set is \([-2,2]\). A direct one-dimensional Vitali selection gives the crude universal bound \(|\{M_c1_A\ge1/3\}|\le9|A|\), hence \(2\le H_3\le9\). No numerical optimization over three-component endpoint types was completed.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `centered-maximal-three-interval-halo`, 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

Reference review is pending for 1 entry. Each affected row names the fields still awaiting review.

1. <a id="reference-1"></a>Exact rational argument and executable replay in cm3ih-artifact-rational-construction https://arxiv.org/abs/math/0311452
   - Also cited at Antonios D. Melas, The best constant for the centered Hardy-Littlewood maximal inequality, Annals of Mathematics 157 (2003), Theorem 1; DOI 10.4007/annals.2003.157.647
   - Also cited at The upper endpoint is Melas's theorem; the lower endpoint is cm3ih-claim-rational-lower-bound
   - Also cited at Inline CPython standard-library computation executed on 2026-07-24
   - preprint; reference source; checked 2026-07-24
   - Reference review pending: publication-style citation, source version.
   - Source use: citation_only
   - Source named by the research packet.