[#P2692] Sharp L2 norm of the centered maximal operator on C_31

Problem. For \(f:\mathbb Z/31\mathbb Z\to\mathbb R\), define \(Mf(j)=\max_{0\leq r\leq15}(2r+1)^{-1}\sum_{k=-r}^{r}|f(j+k)|\). Determine the exact operator norm \(\sup_{f\neq0}\|Mf\|_2/\|f\|_2\).

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1Context

Alternating active-radius and singular-vector updates give a reproducible lower bound 1.3303042705.

2Remarks

Remark 1. Indices are cyclic modulo 31.

Remark 2. The Euclidean norm is unnormalized; the quotient is unchanged by this choice.

3What counts as a solution

Give the exact norm as an algebraic number or certified isolating interval, an extremizing vector, and a complete active-pattern exclusion certificate.

1Status

What counts as a solution

Current status (The L2 norm lies between 1.33030427059916347 and 1.63067915195310467). An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound.^[1] Relevance to this problem For Sharp L2 norm of the centered maximal operator on C_31, record c31m-claim-certified-norm-interval (“The L2 norm lies between 1.33030427059916347 and 1.63067915195310467”) records a bound, answer, status fact, or structural consequence. The record states: An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound. 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 maximal operator over all graph-metric balls of the cycle C_31 Sources Reference 1 Details Let \(N_{31}\) denote the requested norm. Exact rational evaluation of the integer vector in c31m-claim-exact-lower-witness gives \[ \sqrt{\frac{95224622960985697907125617859822805279079620426252}{53808054668649331334697257108564850111138302100375}} \leq N_{31}. \] The lower endpoint is \[ 1.3303042705991634737558623603323047716\ldots. \] The weighted Cauchy-Schwarz argument in c31m-claim-diagonal-upper-certificate proves \[ N_{31}\leq \sqrt{\frac{1916477}{720720}} =1.6306791519531046652193954473854436190\ldots. \] Thus the candidate's reported value is a certified lower bound to ten decimal places. This entry leaves the exact norm and the complete active-pattern exclusion open. Open full record →
All active radii and the squared norm quotient are verified with rational arithmetic.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2 All active radii and the squared norm quotient are verified with rational arithmetic.^[1] Relevance to this problem For Sharp L2 norm of the centered maximal operator on C_31, record c31m-claim-exact-lower-witness (“An exact integer witness attains ratio 1.3303042705991634737...”) records a bound, answer, status fact, or structural consequence. The record states: All active radii and the squared norm quotient are verified with rational arithmetic. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope one explicit nonnegative integer vector on C_31, with every one of its 496 centered averages compared exactly Sources Reference 1 Details In cyclic coordinate order, take \[ \begin{aligned} x={}&(60240784093944,71756706270677,71756706270677,69429053137658,\\ &69429053137658,124946197568962,124946197568962,634350542282289,\\ &220384421138340,634350542282289,124946197568962,124946197568962,\\ &69429053137658,69429053137658,71756706270677,71756706270677,\\ &60240784093944,60240784093944,53164211021439,53164211021439,\\ &49020565351307,49020565351307,47095643667602,47095643667602,\\ &47095643667602,47095643667602,49020565351307,49020565351307,\\ &53164211021439,53164211021439,60240784093944). \end{aligned} \] Exact comparison of the sixteen averages at every coordinate selects the radii \[ (9,8,7,6,5,4,1,0,1,0,1,4,5,6,7,8,9,10,11,12,13,15,15,15,15,15,15,13,12,11,10). \] There are no ties. Direct rational summation gives \[ \frac{\\|Mx\\|_2^2}{\\|x\\|_2^2} =\frac{95224622960985697907125617859822805279079620426252}{53808054668649331334697257108564850111138302100375}. \] This quotient exceeds \((1.3303042705)^2\) by exact integer comparison. Related: Computation 1 Open full record →
A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 3 A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle.^[1] Relevance to this problem For Sharp L2 norm of the centered maximal operator on C_31, record c31m-claim-diagonal-upper-certificate (“A weighted Cauchy-Schwarz certificate gives N_31 squared at most 1916477/720720”) records a bound, answer, status fact, or structural consequence. The record states: A single radial weight sequence majorizes every centered averaging row and remains diagonal after summing over the cycle. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope every real-valued function on C_31 Sources Reference 1 Details Replace \(f\) by \(x=\|f\|\), which preserves the input norm and the maximal function. Set \[ d_0=1,\qquad d_k=d_{-k}=\frac1{4k}\quad(1\leq k\leq15). \] For a radius \(r\), the exact identity \[ \sum_{k=-r}^{r}\frac{1}{(2r+1)^2d_k} =\frac{1+8\sum_{k=1}^{r}k}{(2r+1)^2}=1 \] turns weighted Cauchy-Schwarz into \[ \left(\frac1{2r+1}\sum_{k=-r}^{r}x_{j+k}\right)^2 \leq \sum_{k=-r}^{r}d_kx_{j+k}^2 \leq \sum_{k=-15}^{15}d_kx_{j+k}^2. \] The right side is independent of the selected radius, so it also bounds \((Mx(j))^2\). Summing over all cyclic centers makes every input coordinate receive the coefficient \[ d_0+2\sum_{k=1}^{15}d_k =1+\frac12H_{15} =\frac{1916477}{720720}. \] Therefore \(\\|Mx\\|_2^2\leq(1916477/720720)\\|x\\|_2^2\) for every input. The artifact checks all sixteen coefficient identities over the rationals. Related: Computation 1 Open full record →
Gonzalez-Riquelme and Madrid study the same graph-ball operator and derive exact L2 norms for complete and star graphs.	Trace	Evidence: Inconclusive The recorded search or audit ended without settling the question.
attempt · Route 1 Gonzalez-Riquelme and Madrid study the same graph-ball operator and derive exact L2 norms for complete and star graphs.^[1] Relevance to this problem For Sharp L2 norm of the centered maximal operator on C_31, record c31m-attempt-primary-literature-check (“Finite-graph norm literature gives context but no C_31 value”) documents a concrete method, search boundary, or failed route. The record states: Gonzalez-Riquelme and Madrid study the same graph-ball operator and derive exact L2 norms for complete and star graphs. Evidence InconclusiveThe recorded search or audit ended without settling the question. Scope primary literature located for exact L2 norms of centered Hardy-Littlewood maximal operators on finite graphs, with attention to the cycle C_31 Sources Reference 1 What happened Gonzalez-Riquelme and Madrid define the centered maximal operator on a finite graph by taking averages over graph-metric balls. This specializes to the operator in the candidate record when the graph is \(C_{31}\). Their L2 results give exact formulas for complete graphs and star graphs. The paper supplies no corresponding cycle formula. A focused search by the cycle notation, graph name, and operator norm found nearby variation results, while the exact \(C_{31}\) constant remained unlocated. Novelty therefore remains unverified. Related: Computation 1 Open full record →
The standard-library program uses Fraction for every average, quotient, and certificate identity, then Decimal only for display.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 The standard-library program uses Fraction for every average, quotient, and certificate identity, then Decimal only for display. Relevance to this problem For Sharp L2 norm of the centered maximal operator on C_31, record c31m-artifact-exact-replay (“Exact lower-witness and upper-certificate replay”) supplies evidence or a replay used to check the packet. The record states: The standard-library program uses Fraction for every average, quotient, and certificate identity, then Decimal only for display. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope all 496 averages of the displayed C_31 witness and all sixteen weighted upper-certificate identities Entry point `Join source_lines with LF characters and execute the resulting Python program` Runtime CPython 3.9 or later, standard library only Details The replay evaluates each of the sixteen centered averages at each of the 31 coordinates. It verifies the active-radius vector, reduces the squared quotient to its exact fraction, and proves the stated decimal lower bound by cross multiplication. For the upper endpoint it checks \(\sum_{\|k\|\leq r}((2r+1)^2d_k)^{-1}=1\) at every radius. It then sums the radial weights and obtains \(1916477/720720\). The witness together with its active radii has SHA-256 digest `d6e9301b3a7dc898333a7bb1c2d70a9516db17e08bc9650ea0830d128c9f5cab`. The six-line output has SHA-256 digest `5b55424e8e5e369a4b7034305fc525ae05286e9d9cd7c6f049841ecd565ca8b4`. Related: Computation 2 · Computation 3 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-25. An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound. The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: An exact integer witness reproduces the candidate lower bound, while a closed-form diagonal certificate supplies a universal upper bound.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. One locally consistent active-radius pattern is [9,8,7,6,5,4,1,0,1,0,1,4,5,6,7,8,9,10,11,12,13,15,15,15,15,15,15,13,12,11,10].

Computational notes

Five hundred seeded active-set iterations found ratio 1.3303042705991635. Recomputing all 16 centered averages at every coordinate verified the displayed active pattern, and direct singular-value iteration reproduced the ratio.

How the 5 records connect

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

ProblemSharp L2 norm of the centered maximal operator on C_31

All 5 recorded relations between these records and the problem

2See also

Flattest 32-term Littlewood polynomial on the unit circleharmonic analysis
Sharp fourth-power norm of the cyclic Hilbert transform at order 31harmonic analysis
Openness of convolution on l1 of the integersharmonic analysis

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“Sharp L2 norm of the centered maximal operator on C_31.” TheoremDB. P2692. Problem statement; statement text SHA-256 fee94bc63c75d96700b116dbce7a9f29ff2294a3a762e8f0db3fe6e7352ba1fb. https://theoremdb.org/statement/?ref=P2692

BibTeX

@misc{theoremdb-problem-fee94bc63c75d96700b116dbce7a9f29ff2294a3a762e8f0db3fe6e7352ba1fb,
  title = {{Sharp L2 norm of the centered maximal operator on C\_31}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 fee94bc63c75d96700b116dbce7a9f29ff2294a3a762e8f0db3fe6e7352ba1fb},
  url = {https://theoremdb.org/statement/?ref=P2692}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 5 typed links, sourced from arxiv.org^[1], current as of July 25, 2026.

1Lean verification

1References

Packet source. Finite-graph norm literature gives context but no C_31 value. Cristian Gonzalez-Riquelme and Jose Madrid, Sharp inequalities for maximal operators on finite graphs, arXiv:2005.03146, definition (1.1) and section 1.3. ↗preprint · primary source · arXiv:2005.03146, version checked 2026-07-25 · checked 2026-07-25Source use: original summary.Finite-graph norm literature gives context but no C_31 value. Gonzalez-Riquelme and Madrid study the same graph-ball operator and derive exact L2 norms for complete and star graphs.Also cited at Exact standard-library replay in c31m-artifact-exact-replay.Also cited at The integer witness and all active-radius comparisons are replayed in c31m-artifact-exact-replay.Also cited at Symbolic proof in this record and exact coefficient replay in c31m-artifact-exact-replay.Source named by the research packet.

Original CC0 sharp-constant problem for a finite maximal operator.

[#P2692] Sharp L2 norm of the centered maximal operator on C_31

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