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

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 \(H_3=\sup_A |\{x:M_c1_A(x)\ge1/3\}|/|A|\).

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Definitions and notation

1Context

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

2Conventions

Convention 1. Intervals may touch, and zero-length components may be discarded.

Convention 2. Vertical bars denote Lebesgue measure.

3What 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.

1Status

Saved packet · July 24, 2026

What counts as a solution

Packet-reported status (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.^[1]

This research-packet status has not passed editorial review.

1Packet records

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Record	Kind	Assessment
The packet certifies \(61/20\le H_3\le(11+\sqrt{61})/4\); the exact value of \(H_3\) remains open.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 The packet certifies \(61/20\le H_3\le(11+\sqrt{61})/4\); the exact value of \(H_3\) remains open.^[1] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope the centered Hardy-Littlewood halo at threshold 1/3 for one explicit union of three bounded intervals Sources Reference 1 Details 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. Related: Route 1 Open full record →
The sharp weak (1,1) constant for the centered maximal operator yields the rigorous upper bound 4.702562418976664.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 The sharp weak (1,1) constant for the centered maximal operator yields the rigorous upper bound 4.702562418976664.^[1] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope all measurable subsets of the real line of finite positive measure, including unions of at most three bounded intervals Sources Reference 1 Argument Melas proved that the one-dimensional centered Hardy-Littlewood maximal operator has sharp weak \((1,1)\) constant \[ C=\frac{11+\sqrt{61}}{12}. \] Apply the inequality to \(f=\mathbf 1_A\). For every \(t<1/3\), \[ \|\{M_c\mathbf 1_A\geq1/3\}\| \leq \|\{M_c\mathbf 1_A>t\}\| \leq \frac{C}{t}\|A\|. \] Letting \(t\) increase to \(1/3\) gives \[ H_3\leq 3C=\frac{11+\sqrt{61}}4 =4.702562418976663\ldots. \] The theorem applies to every integrable function, so the restriction to three intervals needs no extra hypothesis. Related: Route 1 Open full record →
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.	Trace	Outcome: Open Evidence: Conjectured Proposed as likely true. No argument is recorded.
attempt · Route 1 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] Relevance to this problem This records a line of attack on the target problem, including what it establishes and where it stops. Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope all unions of at most three bounded intervals at threshold 1/3 Sources Reference 1 What happened The search reduces each rational test set to halo intervals associated with consecutive blocks of the three components. It found the exact construction recorded above. A proof of optimality would have to control every overlap order among the six candidate block intervals, including cases where the full halo has several components. That case analysis has not been completed. The present entry therefore records \[ \frac{61}{20}\leq H_3\leq\frac{11+\sqrt{61}}4 \] as rigorous bounds and leaves the exact constant open. Open full record →
A standard-library Python script computes all six block halos, merges them, and checks the ratio with Fraction arithmetic.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 A standard-library Python script computes all six block halos, merges them, and checks the ratio with Fraction arithmetic.^[1] Relevance to this problem This supplies a computation, dataset, or executable check used to assess the target problem. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the six consecutive-component blocks of the displayed three-interval set Sources Reference 1 Entry point `join source_lines with newline and run with python3` Runtime CPython 3.9 or later Details The replay uses exact rational endpoints. For each consecutive block it constructs the interval of centers that can cover the block within radius \(3m/2\). It checks that all six intervals join into \([-1/10,59/20]\), while the original set has measure 1. Related: Computation 1 Open full record →
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Notes and companion material

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

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.

How the 4 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemA centered maximal-function halo constant for three intervals

All 3 recorded relations between these records and the problem

2See also

Sharp long-shift autocorrelation for half-measure circle setsmeasure theory
The value of Levine's two-player coin-index gamemeasure theory
Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measuremeasure theory

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Plain text

“A centered maximal-function halo constant for three intervals.” TheoremDB. P2546. Problem statement; statement text SHA-256 11f9630e36e8a8479ba7ba3b84d497c05787ee6bd9e8f4dcc3e34e74d88c92c3. https://theoremdb.org/statement/?ref=P2546

BibTeX

@misc{theoremdb-problem-11f9630e36e8a8479ba7ba3b84d497c05787ee6bd9e8f4dcc3e34e74d88c92c3,
  title = {{A centered maximal-function halo constant for three intervals}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 11f9630e36e8a8479ba7ba3b84d497c05787ee6bd9e8f4dcc3e34e74d88c92c3},
  url = {https://theoremdb.org/statement/?ref=P2546}
}

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This problem includes 4 records joined by 3 typed links, sourced from arxiv.org^[1], current as of July 24, 2026.

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Packet source. Exact rational argument and executable replay in cm3ih-artifact-rational-construction. ↗preprint · reference source · checked 2026-07-24Reference review pending: publication-style citation, source version.Source use: citation only.Source named by the research packet.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.

Original finite-component sharp halo problem for the centered Hardy-Littlewood maximal operator.

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

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