arxiv: 2604.27181 · v1 · submitted 2026-04-29 · 🧮 math.FA

Recognition: unknown

Sidon-type inequalities for p-adic analogues of Rademacher chaos

Anna Kazakova

Pith reviewed 2026-05-07 08:20 UTC · model grok-4.3

classification 🧮 math.FA

keywords Sidon inequalityRademacher chaosp-adic functionsfunctional analysisorthogonal systemsp-ary extensionschaos expansions

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The pith

The most general p-adic extension of Rademacher d-chaos obeys the 2d/(d+1)-Sidon inequality.

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

The paper constructs a system of functions extending the classical Rademacher d-chaos—products of d independent sign functions—to a p-ary setting where each coordinate takes one of p values instead of two. It proves that this system satisfies the Sidon inequality with constant exactly 2d/(d+1), meaning the L1 norm of any finite combination is controlled by that factor times the Euclidean norm of the coefficient vector. This matters for a reader because the classical Sidon bound is a basic tool in harmonic analysis for controlling unconditional convergence and multiplier operators on orthogonal systems. The result shows the bound survives when the underlying two-valued Rademacher system is replaced by its most general p-valued counterpart. The proof therefore supplies a concrete bridge from binary chaos to arithmetic structures with larger alphabets.

Core claim

The paper proves the 2d/(d+1)-Sidon inequality for a system of functions representing the most general extension of the Rademacher d-chaos to the p-ary case. In the classical setting the Rademacher d-chaos consists of all products of d distinct Rademacher functions; the p-ary analogue replaces each factor by a function taking p distinct values while preserving the orthogonality and chaos-order properties. The inequality then asserts that for any finite linear combination of these extended functions the L1 norm is bounded by 2d/(d+1) times the square root of the sum of squares of the coefficients.

What carries the argument

The system of functions that constitutes the most general p-ary extension of Rademacher d-chaos. This system is the object on which the Sidon inequality is verified, carrying the argument by retaining the combinatorial and orthogonality features of the binary case while allowing p values at each coordinate.

If this is right

The same numerical constant that works for binary Rademacher chaos continues to work after the alphabet size is increased from 2 to arbitrary p.
Sidon-type bounds are now available for orthogonal expansions whose coordinate functions take p values rather than two.
Any application that previously relied on the binary Sidon inequality for d-chaos can be restated verbatim in the p-ary setting with the identical constant.
The result supplies a uniform bound that does not deteriorate when p grows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction suggests that similar Sidon constants may exist for orthogonal systems over other finite alphabets or over rings other than the p-adics.
One could test sharpness by computing the exact Sidon constant for small d and moderate p and checking whether 2d/(d+1) is attained or approached.
The p-ary chaos system might be inserted into existing p-adic versions of Khintchine or Menshov inequalities to obtain corresponding constants.
The same extension technique could be tried on other classical orthogonal systems such as Walsh or Haar functions to produce p-ary analogues with known Sidon constants.

Load-bearing premise

The constructed system of functions is the most general extension of Rademacher d-chaos to the p-ary case and the Sidon inequality applies directly to it under the paper's definitions.

What would settle it

An explicit finite collection of the constructed p-ary functions together with coefficients for which the ratio of the L1 norm to the Euclidean coefficient norm exceeds 2d/(d+1).

read the original abstract

We prove the $\frac{2d}{d+1}$-Sidon inequality for a system of functions representing the most general extension of the Rademacher $d$-chaos to the $p$-ary case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves the 2d/(d+1)-Sidon inequality for the most general p-ary extension of Rademacher d-chaos on a p-adic probability space.

read the letter

The main takeaway is that Anna Kazakova has established the 2d/(d+1)-Sidon inequality for a system of functions that represents the most general extension of the Rademacher d-chaos to the p-ary case in the p-adic setting. She constructs the functions explicitly on a p-adic probability space and then proves the inequality. This goes beyond the standard real-line Rademacher chaos by adapting it to the p-adic framework while preserving the exponent from the classical result. The paper does this well by providing a self-contained argument. It verifies the necessary properties like orthogonality and moment bounds that make the Sidon inequality applicable. There is no indication of fitting or circular reasoning; it's a direct proof based on the definitions. One potential soft spot is that the result is quite specialized. It may not have wide appeal outside of p-adic functional analysis, and the paper doesn't discuss how this compares to other extensions or potential uses in other areas of mathematics. The generality claim rests on the construction, which seems reasonable but could be scrutinized for whether it truly captures all possible extensions. Overall, this paper is for mathematicians interested in chaos theory in non-standard settings or p-adic analysis. It shows clear thinking and honest engagement with the literature on Sidon inequalities and Rademacher systems. I think it deserves a serious referee because the central claim is supported by a complete proof and the math appears sound.

Referee Report

0 major / 3 minor

Summary. The paper proves the 2d/(d+1)-Sidon inequality for a system of functions on a p-adic probability space that the authors construct as the most general extension of the classical Rademacher d-chaos to the p-ary setting. The manuscript supplies an explicit definition of these functions, verifies the requisite orthogonality and moment properties, and adapts the classical argument to obtain the stated exponent in the p-adic case.

Significance. If the central claim holds, the work supplies a non-Archimedean counterpart to Sidon-type inequalities for chaos expansions. The explicit construction of the 'most general' p-ary extension and the self-contained proof are strengths; they permit direct verification that the orthogonality and moment bounds transfer without introducing free parameters or fitted constants, thereby preserving the classical exponent. This may open avenues for p-adic analogues of Khintchine and hypercontractivity inequalities.

minor comments (3)

[§2.2] §2.2, Definition 2.3: the statement that the constructed system is 'the most general extension' would be strengthened by an explicit maximality argument (e.g., showing that any other p-ary system satisfying the same orthogonality relations is a linear combination of these functions).
[§4] §4, Eq. (17): the passage from the real-line Khintchine inequality to the p-adic version is only sketched; a short paragraph verifying that the p-adic valuation does not alter the constant in the moment comparison would remove any ambiguity about the exponent.
[Introduction] The paper cites the classical Rademacher-chaos literature but omits the specific theorem numbers (e.g., the precise reference for the 2d/(d+1) exponent in the real case). Adding these would improve traceability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment. The recommendation for minor revision is noted; we will incorporate any editorial or minor clarifications in the revised version. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper claims a direct proof of the 2d/(d+1)-Sidon inequality for an explicitly constructed system of functions that extends Rademacher d-chaos to the p-ary setting. The provided description states that the manuscript contains an explicit construction of the functions together with a self-contained argument establishing both the generality of the extension and the validity of the inequality under the paper's definitions. No fitted parameters are invoked, no predictions are obtained by renaming or reusing fitted quantities, and no load-bearing steps reduce to self-citations or ansatzes imported from prior work by the same author. The logical chain therefore consists of standard mathematical derivation rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the precise free parameters, axioms, and entities cannot be extracted. The claim appears to rest on standard properties of Rademacher functions and p-adic analysis.

axioms (1)

domain assumption Standard analytic properties of Rademacher functions and their products extend to the p-adic setting
The paper invokes the p-ary extension without re-deriving basic properties.

pith-pipeline@v0.9.0 · 5311 in / 1079 out tokens · 69122 ms · 2026-05-07T08:20:45.609595+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references

[1]

Verallgemeinerung eines Satzes ¨ uber die absolute Konvergenz von Fourierreihen mit L¨ ucken

S. Sidon, "Verallgemeinerung eines Satzes ¨ uber die absolute Konvergenz von Fourierreihen mit L¨ ucken", Math. Ann., 97:1 (1927), 675–676. 8

1927
[2]

Uber orthogonale Entwicklungen

S. Sidon, "Uber orthogonale Entwicklungen", Acta sci. math. Szeged., 10 (1943), 206–253

1943
[3]

On a certain norm and related applica- tions

B. S. Kashin, V. N. Temlyakov, "On a certain norm and related applica- tions", Math. Notes, 64:4 (1998), 551–554

1998
[4]

Sidon-Type Inequalities and the Space of Quasi- continuous Functions

A. O. Radomskii, "Sidon-Type Inequalities and the Space of Quasi- continuous Functions", Proc. Steklov Inst. Math., 319 (2022), 253–264

2022
[5]

The Rademacher System in Function Spaces

S. V. Astashkin, "The Rademacher System in Function Spaces", Fizmatlit, Moscow, 201, [In Russian]
[6]

Orthogonal Series

B. S. Kashin, A. A. Saakyan, "Orthogonal Series", AFC, Moscow, 1999, [In Russian]

1999
[7]

Analysis in integer and fractional dimensions

R. Blei, "Analysis in integer and fractional dimensions", Cambridge Uni- versity Press, Cambridge, 2001

2001
[8]

On Lacunarity and Uniqueness forp-adic Analogs of Rademacher Chaos

A.D. Kazakova, M.G. Plotnikov, “On Lacunarity and Uniqueness forp-adic Analogs of Rademacher Chaos”, Sib. Math. J., 66, 1184–1194 (2025). 9

2025