Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

Anselmo Raposo Jr; Daniel M. Pellegrino

Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups

T0 review · 0 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proves that the optimal Bohnenblust–Hille constants on finite cyclic groups grow subexponentially in the interaction order.

desk verdict Subexponential BH constants for interaction order are real; the proof is careful and the only genuine weak spot is a load-bearing cited constant that a referee should verify. read the letter →

arxiv 2608.05366 v1 pith:RUQ5HN53 submitted 2026-08-05 math.FA

Daniel M. Pellegrino , Anselmo Raposo Jr This is my paper · ORCID

classification math.FA MSC 42C1042A1643A4646B28

keywords Bohnenblust-HilleinequalityfinitecyclicgroupsinteractionordersubexponentialgrowthPottshypercontractivitymixed-normpolarizationFourieranalysis

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 aims to show that the optimal constants in a dimension-free Bohnenblust–Hille inequality on products of finite cyclic groups grow subexponentially in the interaction order $d$: at most $\exp(c_q\sqrt{d\log d}+O_q(\sqrt{d/\log d}\,\log\log d))$ for every fixed alphabet size $q\ge 2$. Since every function of total degree at most $d$ has interaction order at most $d$, the same bound applies to the classical total-degree constants, making them $\exp(o(d))$ and settling an open question in [2]. A sympathetic reader should care because this replaces an exponential upper bound with a sharp subexponential rate for a strictly larger class of Fourier spectra, and the proof introduces a reusable recurrence that balances a hypercontractive loss against an exact binomial counting factor.

What carries the argument

The argument is carried by a one-step recurrence valid for $2\le m\le d/2$: $$\mathrm{BH}^{\mathrm{int}}_{d,q}\le \mathrm{BH}^{\mathrm{int}}_{m,q}\left(\frac{m+1}{m-1}\right)^{\gamma_q(d-m)}\binom{2d}{2m},$$ where $\gamma_2=1/2$ and $\gamma_q=q\log(q-1)/(4(q-2))$. It is assembled by homogenizing each Fourier character into a $d$-homogeneous polynomial, splitting the $d$ decorated positions into blocks of sizes $m$ and $d-m$, applying Blei's mixed-norm inequality to separate the blocks, and using Potts hypercontractivity to move an $L^2$ norm into an $L^{p_m}$ norm at cost $((m+1)/(m-1))^{\gamma_q(d-m)}$. The remaining mixed evaluation is bounded by a Bernstein-basis polarization estimate whose coefficient sum equals the exact binomial $\binom{2d}{2m}$. Choosing $m\sim\sqrt{2\gamma_q d/\log d}$ makes the hypercontractive loss and the binomial loss coincide at leading order, giving the constant $c_q=2\sqrt{2\gamma_q}$.

What would settle it

Take $d=4$, $m=2$ and exhaustively search the real and complex polynomials of degree at most $4$ whose second Bernstein coefficient has modulus $1$ for the smallest uniform norm; Lemma 6.1 predicts the norm is at least $\binom42/\binom84=3/35$, so a polynomial with smaller norm would refute the polarization estimate and the recurrence that yields Theorem A.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem A: for every fixed $q\ge 2$, as $d\to\infty$, $$\mathrm{BH}^{\mathrm{int}}_{d,q}\le \exp\left(c_q\sqrt{d\log d}+O_q\!\left(\sqrt{\frac d{\log d}}\log\log d\right)\right),$$ with $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\ge 3$. The inequality is dimension-free, holding for functions on $C_q^N$ whose Fourier characters involve at most $d$ active coordinates, and it implies $\mathrm{BH}^{\mathrm{deg}}_{d,q}=\exp(o(d))$ by the inclusion (1.4). This answers affirmatively the question posed in [2] of whether the total-degree constants are subexponential in the degree.

Load-bearing premise

The load-bearing premise is that the Bernstein coefficient bound imported from [10, Proposition 3.2] has exactly the constant $S_m=\sum 4^j\binom d{2j}\binom{d-2j}{m-j}$, with no extra factor growing in $d$; the identity (6.8) then makes the polarization factor exactly $\binom{2d}{2m}$, and any additional factor would turn the recurrence exponential and destroy Theorem A.

Editorial extensions

If this is right

The total-degree constants satisfy $\mathrm{BH}^{\mathrm{deg}}_{d,q}=\exp(o(d))$ for every fixed $q\ge 2$, resolving the open question from [2].
The subexponential estimate holds for the strictly larger interaction-order class, not just bounded total degree, so the result is stronger than the question required.
For every fixed $q$, the bound improves the previous $(C\log q)^{2d}$ estimate from [2] and, for prime $q$, the earlier $C^{d^2}$ estimate from [16].
The leading constant $c_q$ grows only like $\sqrt{2\log q}$ for large $q$, so enlarging the alphabet has a mild effect on the rate.
The constants are independent of the ambient dimension $N$, preserving the dimension-free nature of the earlier finiteness results.

Reading between the lines

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

The recurrence's exact binomial factor is what makes the growth subexponential; if a similar exact count could be found for other alphabets or for non-abelian groups, the same proof scheme would plausibly transfer, but the paper does not claim this.
Because the hypercontractive exponent $\gamma_q$ enters linearly, any sharper log-Sobolev constant for the Potts semigroup would lower the leading constant $c_q$ directly; this is an implicit route to improvement not explored in the paper.
One could test whether the bound remains subexponential when $q$ grows with $d$: the factor $\sqrt{\log q}$ suggests the two parameters may be coupled, but the paper fixes $q$.
In applications to learning low-degree functions, subexponential Fourier-sum constants would improve query-complexity estimates, and the paper's authors explicitly leave learning applications to future work.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this paper actually resolves the Becker-Klein-Slote-Volberg-Zhang question and goes further: it gets exp(o(d)) for the larger interaction-order class, not just total degree. I checked the main line carefully. The recurrence in Proposition 8.2 is the real new idea, and the exponent bookkeeping around it is sound. Blei exponents, the orbit normalization, the polarization identity, and the final optimization all match. The combinatorial identity (6.8) is correct, and the balance at m approximately sqrt(2*gamma_q*d/log d) genuinely produces the claimed c_q*sqrt(d log d) leading term.

Credit where it is due: the paper is well organized, states its assumptions, and does not hide circularity. The recurrence uses the same constant at smaller order, which is a legitimate induction. The block form of Blei's inequality is cited from a published source, even one co-authored by Pellegrino, and that is not a problem on its own.

The soft spot is exactly what the stress-test note flags: Lemma 6.1 imports the Bernstein-coefficient constant from [10, Proposition 3.2] without proof. That constant feeds directly into the binomial factor (2d choose 2m), and any hidden exponential factor would destroy the subexponential bound. The paper could have devoted a paragraph to proving or at least verifying that quoted constant. This is the single most load-bearing unproved premise. However, it is a published result in Math. Ann. by Defant, Mastylo and Perez, so citing it is standard practice. I would call this a genuine but minor concern: worth a referee check, not a reason for rejection.

One smaller presentation issue: in Section 6.2 the change of variables displays a symmetric 2^{-d} factor, and the relation to the cited proposition is not spelled out. It took me a moment to confirm nothing is lost in the normalization. That is fixable by adding one sentence.

Who benefits: anyone working on Bohnenblust-Hille inequalities, Fourier analysis on finite groups, or hypercontractive methods. The paper is self-contained enough for a functional analyst to read the proof directly, and the result improves both [2] and [9] in a clean, quotable way.

My verdict: send it to peer review. A serious referee should verify the imported Bernstein constant and then likely accept. I would not desk reject this; the core argument is solid and the open-question resolution is real.

Referee Report

0 major / 5 minor

Summary. This paper proves a subexponential bound for the Bohnenblust–Hille constants on finite cyclic groups for the interaction-order class: for fixed q≥2, BHint_{d,q} ≤ exp(c_q sqrt(d log d)+O_q(sqrt(d/log d) log log d)), with c_2=2 and c_q=sqrt(2q log(q-1)/(q-2)) for q≥3. The proof proceeds through a recurrence obtained from homogenization, Potts hypercontractivity, Blei's mixed-norm inequality, orbit counting, and a Bernstein-based polarization estimate. Since bounded total degree implies bounded interaction order, the result also gives BHdeg_{d,q}=exp(o(d)), answering a question of Becker, Klein, Slote, Volberg and Zhang.

Significance. The result is a clear step beyond the existing exponential upper bounds for this class and provides the first subexponential estimate in the interaction-order setting. The paper is careful where it matters: the Blei exponent verification (Section 5), the exact combinatorial identity (6.8), the orbit normalization (Section 7), and the asymptotic optimization in Section 10 all check out. The argument contains no fitted parameters, and the leading constant is explicit. The main external ingredient is the exact Bernstein coefficient estimate in Lemma 6.1, quoted from [10, Prop. 3.2]; I found no sign of circularity or internal inconsistency, but the manuscript would be more robust if that proposition were stated verbatim.

minor comments (5)

[6.2, Lemma 6.1] The proof of (6.3) relies on the quoted [10, Prop. 3.2] without reproducing its statement. Since the exact constant S_m is load-bearing for the subexponential conclusion, I recommend stating the quoted proposition in full (including hypotheses) and explicitly showing that the 2^{-d} factors cancel; this would close the only verifiability gap I see.
[7, Eq. (7.5)] The orbit–stabilizer count should state explicitly that the zero symbol is treated as a repeated value with its own multiplicity; otherwise the reader may wonder how the zeros are counted in |[i]|.
[9, Eq. (9.9)] The factor 2^{1/4} is asserted without explanation; a sentence showing that the orbits of size 1 and 2 contribute different ℓ^{4/3} weights would make the base case of Proposition 9.3 transparent.
[10, Eq. (10.9)] It would be helpful to display the line log d = o(sqrt(d/log d) log log d), since the absorption of O_q(log d) into the final error term is otherwise easy to miss.
[Abstract] The sequence 'Masty\l o' is raw LaTeX and should be typeset as 'Mastyło'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the subexponential bound follows from external inequalities and a self-contained optimization; the only author-overlapping citation is independently supported.

full rationale

The derivation chain is an induction: Proposition 8.1 derives a recurrence for BHint_{d,q} from Blei's mixed-norm inequality (Proposition 5.1), Potts hypercontractivity (Lemma 4.1), orbit counting (Section 7), and the mixed polarization estimate (Lemma 6.2). Proposition 8.2 evaluates the polarization sum by the exact identity Lemma 6.3, and Section 10 optimizes the resulting recurrence at m ~ sqrt(2γ_q d / log d). No parameter is fitted to the target quantity; BHint_{d,q} is defined once by (3.8) and never redefined as its own bound. The recurrence legitimately uses BHint_{m,q} at a smaller order, which is the standard induction structure, not circularity. The most delicate imported input is the Bernstein coefficient estimate in Lemma 6.1, quoted from [10, Proposition 3.2]; this is an external published result whose constant is not derived in the paper, so it is a dependency, but not a self-referential one: [10] is not by the present authors and contains no fitted version of BHint_{d,q}. The only author-overlapping citation is [5] for the block form of Blei's inequality, but Proposition 5.1 is independently stated in [10, Proposition 2.1], the paper verifies the exponent specialization, and the central claim does not reduce to [5] by definition. There are no fitted inputs, no uniqueness assertions imported from the authors' prior work, and no renaming of a known empirical pattern as a new theorem. The conclusion BHdeg_{d,q}=exp(o(d)) follows from the monotone inclusion (1.4), which is not circular. Hence the paper's core derivation is self-contained against external benchmarks, and the circularity score is 0.

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

The proof rests on published theorems (hypercontractivity, log-Sobolev constant, Blei's inequality, Bernstein coefficient bound, Littlewood's inequality) and internal combinatorial counting; no free parameters are fitted to data and no new entities are postulated.

assumptions (5)

standard math Gross hypercontractivity for reversible finite Markov semigroups (Lemma 4.1)
Imported from [12] and [8, Theorem 3.5(ii)]; used to derive the Potts hypercontractive estimate (4.15).
standard math Sharp 2-log-Sobolev constant lambda_q for the q-state Potts semigroup (4.9)
From [8, Corollary A.5] and [13, Equation (5)]; determines gamma_q and hence the exponent c_q.
standard math Blei's block mixed-norm inequality (Proposition 5.1)
Restated from [5, Theorem 2.1] and [10, Proposition 2.1]; the exponent verification is given, the inequality itself is cited.
standard math Bernstein basis coefficient estimate (Lemma 6.1)
The real case is quoted from [10, Proposition 3.2]; this supplies the constant S_m that later becomes binom(2d,2m).
standard math Littlewood's 4/3 inequality for bilinear forms (9.7)
Used in the order-two base estimate Proposition 9.3; universal constant L_Lit.

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Cite this review

Pith. "Pith review of Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups." pith.science (2026). https://pith.science/paper/RUQ5HN53

@misc{pith2026260805366,
  author       = {Pith},
  title        = {Pith review of: Subexponential Bohnenblust-Hille Inequalities on Finite Cyclic Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RUQ5HN53}},
  note         = {Machine review of arXiv:2608.05366}
}

abstract

Slote, Volberg and Zhang proved a dimension-free Bohnenblust--Hille inequality on products of finite cyclic groups for functions of bounded total degree. Later, Becker, Klein, Slote, Volberg and Zhang proved $\BHdeg{d}{q}\leq(C\log q)^{2d}$ and asked whether the optimal constants are subexponential in the degree. More recently, Defant, Galicer, Mansilla, Masty\l o and Muro established an exponential Bohnenblust--Hille estimate for the larger support-sensitive class governed by the number of active coordinates. We prove that the optimal constants for this larger class grow subexponentially. More precisely, if $\BHint{d}{q}$ denotes the optimal constant for functions on $C_q^N$ whose Fourier characters involve at most $d$ coordinates, with no restriction on the nonzero local frequencies, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right) \qquad(d\to\infty), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. Since bounded total degree implies bounded interaction order, this also answers the question of Becker et al. The proof uses Potts hypercontractivity, Blei's mixed-norm inequality, orbit counting, and a mixed polarization estimate.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [2]

Becker, O

L. Becker, O. Klein, J. Slote, A. Volberg and H. Zhang,Dimension-free discretizations of the uniform norm by small product sets, Invent. Math.239(2025), 469–503, DOI: 10.1007/s00222-024-01306-9

doi:10.1007/s00222-024-01306-9 2025

[2] [9]

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

A. Defant, D. Galicer, M. Mansilla, M. Mastyło and S. Muro,Support-sensitive Bohnenblust–Hille inequalities and local invariants on Hamming schemes, arXiv:2607.05594v1 (2026)

work page Pith review arXiv 2026

[3] [1]

Arunachalam, A

S. Arunachalam, A. Dutt, F. Escudero Gutiérrez and C. Palazuelos,A cb-Bohnenblust–Hille inequality with constant one and its applications in learning theory, Math. Ann.392(2025), 3367–3396, DOI: 10.1007/s00208-025-03142-5

doi:10.1007/s00208-025-03142-5 2025

[4] [3]

H. F. Bohnenblust and E. Hille,On the absolute convergence of Dirichlet series, Ann. of Math. (2)32 (1931), 600–622

work page 1931

[5] [4]

R. C. Blei,Fractional Cartesian products of sets, Ann. Inst. Fourier (Grenoble)29(1979), no. 2, 79–105, DOI: 10.5802/aif.744

work page doi:10.5802/aif.744 1979

[6] [5]

Bayart, D

F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda,The Bohr radius of then-dimensional polydisk is equivalent to p (logn)/n, Adv. Math.264(2014), 726–746

work page 2014

[7] [6]

Carando, A

D. Carando, A. Defant and P. Sevilla-Peris,The Bohnenblust–Hille inequality combined with an inequality of Helson, Proc. Amer. Math. Soc.143(2015), no. 12, 5233–5238, DOI: 10.1090/proc/12551

work page doi:10.1090/proc/12551 2015

[8] [7]

J. N. Caro Montoya, D. Núñez-Alarcón and D. M. Serrano Rodríguez,Asymptotic contractivity of the Bohnenblust–Hille inequality for polynomials with few interacting variables, arXiv:2607.20847v1 (2026)

work page Pith review arXiv 2026