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Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes

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Pith's one-line read Support size, not total degree, governs Bohnenblust–Hille growth for spherical polynomials on Hamming schemes, yielding sharp local-invariant asymptotics and better learning sample bounds.

desk verdict Solid support-sensitive BH inequality that cleans up the complexity parameter for spherical levels on Hamming schemes, with sharp asymptotics for Sidon/χ/gl and projection constants that look correctly proved. read the letter →

arxiv 2607.05594 v1 pith:RJLECY44 submitted 2026-07-06 math.FA math.CV

Andreas Defant , Daniel Galicer , Martín Mansilla , Mieczys{l}aw Masty{l}o , Santiago Muro This is my paper · ORCID
classification math.FAmath.CV MSC 46G2546B0768Q3243A4632A0842C10
keywords Bohnenblust–HilleinequalityHammingschemesphericalpolynomialsSidonconstantGordon–LewisprojectionKrawtchouklearningtheory
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

On the q-ary Hamming scheme, low-complexity polynomials can be measured by how many coordinates they actually use (support size) rather than by total degree. The paper proves a Bohnenblust–Hille inequality whose constants grow only with that support size: the interaction order of the variables, not the potentially larger total degree, is the true complexity parameter. That single estimate feeds dimension-free comparisons of Sidon, unconditional-basis and Gordon–Lewis constants, giving matching asymptotics of order c(q)^d (N/d)^{(d−1)/2} for spherical, homogeneous and tetrahedral spaces. The same control improves learning of low-level functions, replacing accuracy exponents that scale with (q−1)d by exponents that scale only with d. Projection constants are identified as L1 norms of kernels; under central-limit scaling they converge to Hermite moments in the spherical case and to circular-complex-Gaussian moments when q≥3. A top-layer principle then shows that spaces of degree or level at most d inherit the same limits from their highest layer.

What carries the argument

Support-sensitive Bohnenblust–Hille inequality (Theorem 2.13): after a Remez transfer from the torus and a random labelled-block decoupling that isolates tetrahedral multilinear forms, the ℓ^{2d/(d+1)} sum of Fourier coefficients is controlled by D_1(q)^d times the supremum norm, with the optimal exponent 2d/(d+1).

What would settle it

Compute the Sidon constant of the level-d spherical space for large N and fixed d,q and check whether it grows like c(q)^d (N/d)^{(d−1)/2}; a strictly smaller power of N would falsify optimality of the exponent 2d/(d+1) and the asymptotic claim.

Watch

Extended reading notes

Core claim

The paper establishes a support-sensitive Bohnenblust–Hille inequality: every function in the level-at-most-d spherical space on C_q^N obeys a Fourier-coefficient bound with optimal exponent 2d/(d+1) and constants D_1(q)^d where D_1(q)=O(√q log q). Support size, not total degree, is the governing complexity parameter. From this it derives matching asymptotics for Sidon, unconditional-basis and Gordon–Lewis constants of the spherical, homogeneous and tetrahedral spaces, together with explicit Gaussian (or circular-Gaussian) limits for their normalized projection constants.

Load-bearing premise

The argument needs the Remez transfer and multilinear Bohnenblust–Hille bounds to grow only exponentially in the support size d (with base depending on q alone); if either constant grew substantially worse, every dimension-free comparison and the learning sample size would degrade.

Editorial extensions

If this is right

  • Sidon, unconditional-basis and Gordon–Lewis constants of B_d, P_d and T_d are all asymptotically equivalent to c(q)^d (N/d)^{(d−1)/2}.
  • Learning low-level functions needs only O(d D_1(q)^{2d^2} ε^{-(d+1)} log(L/δ)) samples, improving the accuracy exponent from (q−1)d+1 to d+1.
  • Normalized projection constants of spherical spaces converge to E|He_d(Z)|/√(d!), while homogeneous and tetrahedral spaces (q≥3) converge to Γ(1+d/2)/√(d!).
  • Spaces of level or degree at most d inherit the same normalized projection-constant limits from their top layer.
  • The same comparisons extend immediately to the cumulative spaces B_≤d, P_≤d and T_≤d.

Reading between the lines

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

  • The same support-sensitive philosophy should apply to other product-group Fourier settings (e.g., non-abelian or non-uniform measures) whenever only the number of active coordinates matters.
  • Once the Remez constant is known more sharply for q≫d, the base c(q) can be driven close to 1, tightening all sample-complexity and invariant estimates in the fine-grid regime.
  • The circular-Gaussian limit for q≥3 suggests that higher-moment projection phenomena on cyclic groups are governed by free circular ensembles rather than real Gaussians.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper develops support-sensitive Bohnenblust–Hille inequalities for spherical (level-d) polynomial spaces on the q-ary Hamming scheme C_q^N, showing that the governing complexity is the support size of monomials rather than total degree (Theorems 2.4, 2.9, 2.13). Combined with a support-tracked Remez transfer from the torus (Theorem 2.2) and dimension-free comparison principles, this yields sharp asymptotics Sidon/χ/gl ∼ c(q)^d (N/d)^{(d-1)/2} for the spherical, homogeneous and tetrahedral spaces (Theorem 3.3 and Corollary 3.6). Projection constants are identified via kernels: normalized Krawtchouk polynomials converge to Hermite polynomials, giving the Gaussian limit E|He_d(Z)|/√(d!) for spherical spaces (Theorem 4.1); for homogeneous/tetrahedral spaces a Boolean/q≥3 dichotomy appears, with the latter governed by circular-complex-Gaussian moments Γ(1+d/2)/√(d!) (Theorem 5.1). Learning sample-complexity bounds for low-level functions improve the ε-dependence from the total-degree regime (Theorem 2.14).

Significance. If correct, the work supplies a clean conceptual advance: interaction order (support size) rather than total degree controls the constants for q-capped polynomials, removing the (q-1)d blow-up that appears in earlier total-degree BH/Remez results. The resulting dimension-free equivalences among Sidon, unconditional-basis and Gordon–Lewis constants, the sharp asymptotic formulae, and the explicit projection-constant limits (Hermite versus circular Gaussian) are of lasting interest in harmonic analysis on product groups, local Banach-space theory and discrete learning. The proofs are fully written, the decoupling constant is shown to be sharp up to absolute factors, and the top-layer principle unifies the cumulative-space asymptotics; these are genuine strengths.

minor comments (5)
  1. In the statement of Theorem 2.13 the constant is written D_1(q)=O(√q log q); the same order appears for D(q) in Theorem 2.9. A single sentence clarifying that the log q factor originates solely from the Remez transfer (Theorem 2.2) while the √q factor comes from the local decoupling (Lemma 2.5) would help the reader track the sources.
  2. Page 36, display after Hölder: the estimate (N choose d) ≤ (eN/d)^d is used without citation; a parenthetical reference to the standard Stirling bound would be useful for non-specialists.
  3. In Proposition 4.5 the uniform-on-compacts convergence of normalized Krawtchouk polynomials is proved via generating functions and Cauchy integrals; the subsequent probabilistic transfer (Theorem 4.4) invokes Slutsky. Adding one sentence that the family is tight by the classical CLT would make the argument self-contained for readers less familiar with weak-convergence tools.
  4. Notation for the cumulative spaces B_{≤d}, P_{≤d}, T_{≤d} is introduced early but the corresponding toric versions B_{≤d,q}(T^N) appear only later; a brief forward reference in §1.1 would improve readability.
  5. A few typographical slips: “dimen-sion” (p. 11), “Steˇ ckin” (p. 15), and the occasional missing space before punctuation in displayed formulae. These are easily cleaned in production.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: support-sensitive BH, local-invariant asymptotics, and projection-constant limits are derived from independent multilinear BH, Remez transfers, and classical orthogonal-polynomial CLTs, not forced by definition or self-fit.

  1. self citation load bearing [Section 3.1, Corollary 3.2 and Theorem 3.1 (citing [10, Theorem 2.5])]
    "The next result, which is a special case of [10, Theorem 2.5], shows that for arbitrary sets of monomials of degree at most m, the Gordon–Lewis constant, the unconditional basis constant, and the monomial unconditional basis constant are equivalent up to the exponential factor 2^m. ... Combining Theorem 3.1 with the Remez-type transfer in Theorem 2.2, and using the ideal property of the Gordon–Lewis constant, we obtain the following comparison on the finite group C_q^N."

    The dimension-free equivalence gl ≤ χ ≤ Sid ≤ (C log q)^{2d} 2^m gl for the spherical/homogeneous/tetrahedral spaces is obtained by specializing the authors’ own prior comparison theorem for bounded-degree monomials on the torus and transferring via Remez. The comparison itself is not circular (it is a general Banach-space fact), but the load-bearing step that converts toroidal monomial unconditionality into discrete Sidon/χ/gl asymptotics rests on this self-citation. The subsequent asymptotic order (N/d)^{(d−1)/2} is still independently supplied by the new support-sensitive BH and by lower bounds from tetrahedral subsystems, so the circularity is minor and non-decisive.

full rationale

The paper’s central claims (Thm 2.13 support-sensitive BH with optimal exponent 2d/(d+1); Thm 3.3 Sidon/χ/gl ∼ c(q)^d (N/d)^{(d−1)/2}; Thms 4.1/5.1 normalized projection constants → E|He_d(Z)|/√(d!) or Γ(1+d/2)/√(d!)) rest on a transparent, non-circular chain. Local decoupling (Lem 2.5) replaces powers by independent variables at cost O(√q); random labelled partitions recover the positive proportion d!/d^d of coefficients (absorbed into the base, not a hidden factorial loss); multilinear BH with known sub-exponential growth is applied; and a support-tracked Remez transfer (Thm 2.2) yields the discrete estimate. Dimension-free comparisons of Sidon/χ/gl use the authors’ prior comparison principle [10] and Remez, but that principle is an independent general statement for bounded-degree monomials, not a restatement of the target asymptotics. Projection constants are identified with L1-norms of character kernels (standard for compact Abelian groups, [9,16,26]), then evaluated via Krawtchouk→Hermite CLT (spherical) or circular-complex-Gaussian moments (homogeneous/tetrahedral, q≥3). No quantity is fitted to data and then “predicted”; no uniqueness theorem is imported solely to forbid alternatives; and the Boolean/q≥3 dichotomy follows from the distinct limiting laws rather than from renaming. Self-citations supply tools with independent content; residual uncertainty lies only in the quality of the external Remez/BH black boxes, which is a correctness risk, not circularity. Score 1 reflects one minor, non-load-bearing self-citation pattern only.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper is pure analysis on finite Abelian groups and the torus. Load-bearing inputs are standard Fourier analysis, known multilinear Bohnenblust–Hille inequalities, Remez-type discretization results from the recent literature, and classical asymptotics of Krawtchouk/Hermite polynomials and CLTs for roots of unity. No free parameters are fitted to data; invented objects are only the named polynomial spaces and kernels, which are standard spans of characters.

assumptions (5)
  • standard math Multilinear Bohnenblust–Hille inequality with at most exponential (in fact polynomial) growth of constants BH_d ≤ B^d
    Invoked in the proof of Theorem 2.4 after decoupling; cited via [11, Prop. 6.18], [2], [13].
  • domain assumption Dimension-free Remez-type comparison ∥P∥_{L^∞(T^N)} ≤ (C log q)^d ∥P∥_{L^∞(C_q^N)} for support-level ≤d polynomials (Theorem 2.2, refined from [17,3,28])
    Transfers toroidal estimates to the Hamming scheme; support-sensitive form is extracted by tracking active coordinates in the randomization of [17].
  • standard math Characters of C_q^N form an orthonormal basis of L^2; projection constants of character subspaces equal L^1-norms of the associated kernels (identity (16) from [9,26])
    Used throughout Sections 4–5 for λ(P_Λ).
  • standard math Central limit theorem for binomial Hamming weight and multivariate CLT for sums of uniform roots of unity (q≥3) yielding circular complex Gaussians
    Drives Krawtchouk→Hermite and character-sum→Γ(1+d/2) limits.
  • standard math Gordon–Lewis inequality gl(X) ≤ χ(X) ≤ Sidon of the character basis, and ideal property of gl under factorization
    Section 3 comparison of local invariants; classical [15].
invented entities (2)
  • Support-sensitive / spherical spaces B_d(C_q^N) and toric analogues B_{d,q}(T^N) independent evidence
    purpose: Isolate Fourier spectrum by support size rather than total degree so BH growth depends on interaction order d
    Standard spans of characters; not new physical entities, but the organizing objects of the paper.
  • Decoupling map U_A replacing z_i^r by independent y_{i,r} independent evidence
    purpose: Convert dependent powers of one coordinate into independent variables at cost O(√q)
    Technical device in Lemma 2.5; norm sharpness via Sidon of {z,...,z^{q−1}}.

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Pith. "Pith review of Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes." pith.science (2026). https://pith.science/paper/RJLECY44

@misc{pith2026260705594,
  author       = {Pith},
  title        = {Pith review of: Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RJLECY44}},
  note         = {Machine review of arXiv:2607.05594}
}
abstract

We investigate local invariants and geometric phenomena for polynomial spaces of low degree on the $q$-ary Hamming scheme $C_q^N$, where $C_q$ denotes the cyclic group of order $q$. Our main analytic tool is a support-sensitive Bohnenblust--Hille inequality for spherical polynomial spaces, showing that the relevant complexity parameter is the support size of the monomials rather than their total degree. Equivalently, in the corresponding toroidal formulation, this leads to estimates for polynomials whose coordinate degrees are bounded by $q-1$, while the growth of the constants is governed by the interaction order of the variables. These inequalities yield applications to the learning theory of spherical low-level functions and also provide the basis for dimension-free comparisons between several classical local invariants, including Sidon constants, unconditional basis constants, and Gordon--Lewis constants. As a consequence, we obtain sharp asymptotic estimates for these invariants in the spherical setting, with analogous comparison and asymptotic results for homogeneous and tetrahedral polynomial spaces. We also study projection constants and the associated reproducing kernels. In the spherical case, suitably normalized Krawtchouk polynomials converge to Hermite polynomials under central-limit scaling, leading to explicit Gaussian limits and sharp asymptotic formulas. By contrast, in the homogeneous and tetrahedral settings a dichotomy appears between the Boolean case and the regime $q\ge3$, where the limiting behaviour is governed by moments of a circular complex Gaussian.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

    math.FA 2026-08 accept novelty 7.0 of 10

    For fixed q, the optimal constants BHint_{d,q} grow like exp(c_q sqrt(d log d)), which is subexponential, answering the question of Becker, Klein, Slote, Volberg and Zhang.

  2. Asymptotic contractivity of the Bohnenblust--Hille inequality for polynomials with few interacting variables

    math.FA 2026-07 accept novelty 6.0 of 10

    The optimal Bohnenblust–Hille constant for m-homogeneous polynomials with support size at most M satisfies K_{m,M} → 1 at rate A_M^{M/m} m^{(M^2-1)/(2m)}.

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