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Stolarsky-Type Inequalities in a Max-Convolution Problem

T0 review · 2 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The max-convolution inequality holds for geometric block sequences with exponent q_m = log(2m+1)/(2 log(m+1)) for every natural number m.

desk verdict Hosle extends the geometric-block case to all m via Stolarsky means and adds the two-nonzero-term case, a direct but partial advance on the BDFKK question. read the letter →

arxiv 2606.07946 v1 pith:GS2ABZCT submitted 2026-06-06 math.CO

Johannes Hosle This is my paper · ORCID
classification math.CO
keywords max-convolutionStolarskymeansgeometricblockssumsetsproductsetsdecreasingsequencesinequalities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that the max-convolution inequality with a carefully chosen exponent q_m holds when both sequences are geometric blocks, meaning they consist of powers of a fixed ratio t in [0,1] up to some index and then zeros. This case was the remaining obstacle after earlier work reduced the general decreasing-sequence version to the geometric-block version via a max-tie analysis. Establishing the inequality for these blocks therefore yields an affirmative answer to the BDFKK question on the sizes of sumsets inside product sets. The argument proceeds by comparing Stolarsky means of the terms that appear in the max-convolution.

What carries the argument

Comparison of Stolarsky means applied to the powered terms that arise from the max-convolution of two geometric blocks.

What would settle it

Explicit numerical values of m, r, s and t in [0,1] for which the left-hand sum of the q_m-powers of the max terms falls below the right-hand product of the two q_m-powers of the sums.

Watch

Extended reading notes

Core claim

For every natural number m the inequality sum over k of (max_{i+j=k} x_i y_j)^{q_m} is at least (sum x_i)^{q_m} (sum y_j)^{q_m} when x and y are geometric blocks with common ratio t in [0,1]. The proof is obtained by a direct comparison of Stolarsky means. The same inequality is also verified when one sequence has only two nonzero terms and for certain perturbations of the geometric blocks.

Load-bearing premise

The specific exponent q_m makes a Stolarsky-mean comparison sufficient to prove the inequality for every geometric block without extra restrictions on the block lengths or the ratio t.

Editorial extensions

If this is right

  • The BDFKK question on sumset sizes receives an affirmative answer.
  • The inequality holds when one of the sequences has exactly two nonzero terms.
  • Certain small perturbations of geometric blocks continue to satisfy the inequality.
  • The earlier reduction of the general decreasing-sequence case to the geometric-block case is now justified for this exponent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Stolarsky comparison can be extended beyond pure geometric blocks, the inequality may hold for all monotone sequences.
  • The same mean-comparison technique might apply to other convolution inequalities that appear in additive combinatorics.
  • Direct computation for small m and random t could quickly locate any counterexamples if the claim is false.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The manuscript proves the max-convolution inequality for geometric-block sequences x=(1,t,...,t^r,0,...,0) and y=(1,t,...,t^s,0,...,0) with t in [0,1] and arbitrary natural m, by reducing to a comparison of Stolarsky means with the explicit exponent q_m = log(2m+1)/(2 log(m+1)). This is claimed to yield an affirmative answer to the BDFKK question on sumset sizes in product sets. The paper also verifies some perturbations of the geometric case and proves the inequality when one sequence has exactly two nonzero terms.

Significance. If the central argument holds, the work supplies the missing geometric-block case needed to affirm the BDFKK question, extending the m=2 result of BIKM. The explicit reduction to Stolarsky means with a parameter-free exponent q_m is a clear methodological strength and supplies a falsifiable, checkable criterion for the inequality on geometric sequences.

major comments (2)
  1. [Geometric block case] Geometric-block section: the Stolarsky-mean comparison is presented as sufficient for all r,s, yet when min(r,s) is substantially smaller than m the number of active maximizing pairs drops below 2m+1 and the locations of the maxima shift; the manuscript must supply explicit case distinctions or an auxiliary argument showing that the same q_m still dominates without further restrictions on the support sizes.
  2. [Definition of q_m and Stolarsky comparison] The choice of q_m is asserted to make the two-term Stolarsky comparison control the full (2m+1)-term sum; an explicit derivation or inequality chain showing why this particular logarithmic ratio is the threshold value (rather than a larger or smaller exponent) is required to confirm that the comparison is tight and covers the claimed range of r and s.
minor comments (1)
  1. [Abstract] The abstract states the inequality for arbitrary decreasing sequences but the body restricts to geometric blocks; a short clarifying sentence on the reduction steps from the general case (as done for m=2 in BIKM) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. The two major comments identify points where the presentation can be strengthened with additional case analysis and an explicit derivation of the exponent. We address each below and will incorporate the necessary clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Geometric block case] Geometric-block section: the Stolarsky-mean comparison is presented as sufficient for all r,s, yet when min(r,s) is substantially smaller than m the number of active maximizing pairs drops below 2m+1 and the locations of the maxima shift; the manuscript must supply explicit case distinctions or an auxiliary argument showing that the same q_m still dominates without further restrictions on the support sizes.

    Authors: We agree that the argument as written focuses on the regime where the supports of x and y are large enough to produce 2m+1 distinct maximizing pairs. When min(r,s) is small relative to m the effective convolution length is shorter. In the revision we will insert a preliminary reduction: if min(r,s) = k < m then the geometric-block inequality for parameters (m,r,s) reduces to the same inequality for parameters (k,r,s) together with a comparison of the exponents q_m and q_k. Because q_m is decreasing in m, the smaller exponent q_m yields a weaker (but still valid) lower bound once the k-case has been established; the required auxiliary comparison between the two Stolarsky means will be supplied explicitly. revision: yes

  2. Referee: [Definition of q_m and Stolarsky comparison] The choice of q_m is asserted to make the two-term Stolarsky comparison control the full (2m+1)-term sum; an explicit derivation or inequality chain showing why this particular logarithmic ratio is the threshold value (rather than a larger or smaller exponent) is required to confirm that the comparison is tight and covers the claimed range of r and s.

    Authors: The exponent q_m is the unique value that equates the two-term Stolarsky mean of order q with the (2m+1)-term arithmetic mean in the critical geometric case r = s = m. We will add a short subsection that derives this relation by solving the equality condition for the Stolarsky mean M_q(a,b) applied to the pairs that realize the maximum convolution entries, then verifies by direct computation that the resulting q_m satisfies the required monotonicity and comparison inequalities for all admissible r,s. The chain will be written out in full so that the threshold character of the logarithmic ratio is transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; explicit exponent and external Stolarsky comparison

full rationale

The derivation defines q_m explicitly as log(2m+1)/(2 log(m+1)) and reduces the geometric-block case to a direct comparison of existing Stolarsky means. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation appears in the provided text. The argument is presented as an independent verification for the block sequences, with the m=2 case cited externally to prior work by different authors. This is the normal non-circular outcome for a paper whose central step is an explicit mean comparison rather than a fit or renaming.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The claim rests on the validity of Stolarsky-mean inequalities for geometric sequences with the given exponent; no free parameters are introduced beyond the explicit definition of q_m.

assumptions (1)
  • standard math Stolarsky means satisfy the comparison inequalities needed for the geometric-block case
    Invoked to prove the max-convolution bound for x and y of the stated geometric form.

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Cite this review

Pith. "Pith review of Stolarsky-Type Inequalities in a Max-Convolution Problem." pith.science (2026). https://pith.science/paper/GS2ABZCT

@misc{pith2026260607946,
  author       = {Pith},
  title        = {Pith review of: Stolarsky-Type Inequalities in a Max-Convolution Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GS2ABZCT}},
  note         = {Machine review of arXiv:2606.07946}
}
abstract

For $m \in \mathbb{N}$, let $q_m := \frac{\log(2m+1)}{2\log(m+1)}$. The max-convolution inequality \begin{align*} \sum_{k=0}^{2m}\left(\max_{i+j=k} x_i y_j \right)^{q_m} &\ge \left(\sum_{i=0}^{m} x_i\right)^{q_m} \left(\sum_{j=0}^{m} y_j\right)^{q_m} \end{align*}for arbitrary sequences $x_0 \ge x_1 \ge ... \ge x_m \ge 0, y_0 \ge y_1 \ge ... \ge y_m \ge 0$ implies an affirmative answer to a question of Bourgain, Dilworth, Ford, Konyagin, and Kutzarova \cite{BDFKK} on the sizes of sumsets in product sets. This inequality was proven for $m = 2$ by Becker, Ivanisvili, Krachun, and Madrid \cite{BIKM} by reducing the general case to the geometric block case via a max-tie analysis. We prove the geometric block case $x = (1, t, ..., t^{r}, 0, ..., 0)$ and $y = (1, t, ..., t^s, 0, ..., 0)$, $t \in [0, 1]$, for all $m \in \mathbb{N}$ via a comparison of Stolarsky means. Some perturbations are also verified. Finally, we prove the above inequality when one sequence has only two non-zero terms.

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Forward citations

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Works this paper leans on

7 extracted references · 7 canonical work pages · cited by 1 Pith paper

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