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

Abstract: We investigate local invariants and geometric phenomena for polynomial spaces of low degree on the $q$-ary Hamming scheme $C_q<sup>N$, 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.

• The paper introduces a novel support-sensitive BH inequality that replaces total degree with support size to achieve dimension-free estimates on q-ary Hamming schemes.
• It establishes sharp asymptotic behaviors for projection constants and equivalences between Banach space invariants like Sidon and Gordon–Lewis constants.
• The work provides improved sample complexity bounds for learning low-level functions, enhancing efficiency in high-dimensional inference tasks.

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

Introduction and Context

This paper develops a comprehensive analytical framework for the study of low-degree function spaces on $q$-ary Hamming schemes $C_q^N$, where $C_q$ is the finite cyclic group of order $q$. The interaction between harmonic analysis, probabilistic combinatorics, and local Banach space theory is carefully examined through the lens of support-sensitive inequalities, local invariants such as Sidon and Gordon–Lewis constants, and the structural and asymptotic geometry of associated polynomial spaces.

The central innovation is a new class of Bohnenblust–Hille (BH) inequalities sensitive to the support size of monomials, contrasting with traditional degrees in polynomial bounds. This paradigm shift enables improved dimension-independent estimates—which are then leveraged to advance the theory of learning low-level functions in high-dimensional regimes, as well as to tightly characterize several classical bases and projection constants in functional analysis.

Main Analytical Results

Spherical, Homogeneous, and Tetrahedral Polynomial Spaces

The study focuses on three families of subspaces of $L^\infty(C_q^N)$:

• Spherical spaces $\mathcal{B}_d(C_q^N)$: Functions whose Fourier support is restricted to characters depending nontrivially on exactly $d$ coordinates.
• Homogeneous spaces $\mathcal{P}_d(C_q^N)$: Characters of total degree $d$, corresponding to homogeneous polynomials of degree $d$.
• Tetrahedral spaces $C_q^N$0: Intersection of spherical and homogeneous, i.e., degree $C_q^N$1 and dependence on exactly $C_q^N$2 coordinates, capturing square-free homogeneous monomials.

These structures are critically distinct for $C_q^N$3, exhibiting divergent asymptotic behavior as $C_q^N$4.

Support-Sensitive Bohnenblust–Hille Inequalities

The principal technical contribution is the derivation of a support-sensitive BH inequality for polynomials on $C_q^N$5 whose monomials have support of size at most $C_q^N$6. Formally, for $C_q^N$7,

$C_q^N$8

where $C_q^N$9. The exponent $C_q$0 is sharp. This result replaces total degree as the complexity parameter with support size (interaction order), which is crucial in high-dimensional settings where the total degree can be much larger than the number of active variables.

The proof requires a combination of Remez-type transfer principles, decoupling/block randomization, and avoiding factorial explosion in constants via support-based combinatorics. The approach generalizes and strengthens previous work restricted to total degree bounds [Slote, Volberg, Zhang, 2024; Becker et al., 2025].

Dimension-Free Norm Comparisons and Local Invariants

Utilizing these inequalities, the paper provides dimension-free equivalences between Sidon constants, unconditional basis constants, and Gordon–Lewis constants on these spaces. For $C_q$1 and $C_q$2, for any of the three invariants $C_q$3 and any of the spaces considered,

$C_q$4

where $C_q$5 depends only on $C_q$6. The invariants thus scale sub-exponentially in $C_q$7 for fixed $C_q$8, supporting efficient analysis in high-dimensional function spaces.

Projection Constants and Gaussian Limit Laws

The paper establishes the asymptotic behavior of projection constants $C_q$9 for these spaces as $q$0, revealing fundamentally different limiting behaviors:

• Spherical case: After normalization by $q$1, the projection constant converges to a Gaussian moment involving Hermite polynomials:

$q$2

where $q$3 and $q$4 is the $q$5-th probabilists' Hermite polynomial.

• Homogeneous and Tetrahedral cases ($q$6): The normalized constants converge to moments involving a standard complex Gaussian:

$q$7

reflecting cancellation phenomena in character sums.

• Top-Layer Principle: For spaces corresponding to degree/level at most $q$8, the asymptotics are dominated by the top level ($q$9), rendering lower-degree components negligible in the large $L^\infty(C_q^N)$0 limit.

These results elucidate the connection between the discrete structure of polynomial spaces on $L^\infty(C_q^N)$1 and classical objects in probability and analysis.

Applications to Learning Theory

A central application concerns the statistical learning of low-level functions on Hamming schemes. The results provide improved sample complexity estimates for reconstructing $L^\infty(C_q^N)$2 from random samples:

$L^\infty(C_q^N)$3

guarantees $L^\infty(C_q^N)$4-approximation error $L^\infty(C_q^N)$5 with probability $L^\infty(C_q^N)$6. Support-sensitivity yields exponential improvements in the dependence on accuracy parameter $L^\infty(C_q^N)$7, compared to previous degree-based approaches.

These learning-theoretic consequences are especially relevant to high-dimensional distribution testing, learning Boolean and cyclic functions, and quantum classical query complexity separation.

Theoretical and Practical Implications

Theoretical Implications

• Demonstrates that support size—rather than total degree—is the natural notion of complexity governing local analysis, unconditionality, and norm/learning inequalities in high-dimensional group settings.
• Provides new tools and limits for the geometry of polynomial function spaces, enabling precise asymptotics and equivalence of classical Banach invariants beyond previous dimensional and complexity constraints.
• Connects discrete orthogonal polynomials (Krawtchouk) to Hermite/complex Gaussian regimes, explaining universal Gaussian limits in projection constants.

Practical and Future Directions

• Enables the design of more efficient learning algorithms and uniform approximations for classes of functions with bounded interaction order, impacting property testing and high-dimensional inference on finite groups.
• Motivates the use of support-restricted polynomials as bases for function estimation, with sharper guarantees for sample complexity and regularization in computational settings.
• Suggests avenues for further development of support-sensitive inequalities in other combinatorial, analytic, and computational domains where variable sparsity is structurally critical.
• The sharp asymptotic results for invariants strengthen the bridge between high-dimensional harmonic analysis and functional-analytic geometry, providing technical leverage for studying measures of complexity in other algebraic and combinatorial models.

Conclusion

This work establishes a comprehensive and sharp analytic and asymptotic theory for low-level function spaces on $L^\infty(C_q^N)$8-ary Hamming schemes via support-sensitive Bohnenblust–Hille inequalities. The findings yield powerful dimension-independent estimates for Banach space invariants, new universal limit laws for projection constants, and improved learning-theoretic upper bounds. The support-centric approach not only improves upon degree-based frameworks but also provides structural and practical insights with broad mathematical and algorithmic significance.

References: See the bibliography in "Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes" (2607.05594).