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arxiv: 2604.20774 · v1 · submitted 2026-04-22 · 🧮 math.OA · math.FA

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Finite Riesz products and Ornstein non-inequalities on quantum tori

Christos P. Tantalakis, Micha{\l} Wojciechowski

Pith reviewed 2026-05-09 22:29 UTC · model grok-4.3

classification 🧮 math.OA math.FA
keywords quantum torusRiesz productsOrnstein non-inequalitynon-commutative analysisoperator algebrasharmonic analysisfinite products
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0 comments X

The pith

A construction generalizing finite Riesz products to the quantum torus proves a non-commutative Ornstein non-inequality.

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

The paper shows how to define products on the quantum torus that extend the classical finite Riesz products from the commutative torus. This extension is presented as a natural non-commutative counterpart. By combining the new construction with earlier results on non-inequalities, the authors establish that an Ornstein-type non-inequality holds in this quantum setting. Readers interested in operator algebras would care because it transfers tools from classical analysis to the non-commutative world, where many physical and mathematical models live.

Core claim

We demonstrate a construction of products on the quantum torus T_θ² that generalises the usual construction of finite Riesz products on the commutative torus T². We explain why the former constitutes a natural analogue of the latter in the non-commutative setting and, based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality.

What carries the argument

The generalized finite Riesz product construction on the quantum torus T_θ², which serves as the non-commutative analogue enabling the proof of the Ornstein non-inequality.

If this is right

  • The Ornstein non-inequality extends to the non-commutative quantum torus setting.
  • The construction provides a basis for further harmonic analysis on quantum tori.
  • Classical results on Riesz products carry over when the appropriate non-commutative version is used.

Where Pith is reading between the lines

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

  • This approach could inspire similar generalizations for other inequalities or structures in non-commutative harmonic analysis.
  • Applications might arise in studying dynamical systems or ergodic theory on quantum tori.
  • Testing the construction for specific irrational θ values could reveal new behaviors not present in the commutative case.

Load-bearing premise

The products defined on the quantum torus act as a direct natural analogue to commutative finite Riesz products so that prior results on Ornstein non-inequalities apply immediately.

What would settle it

Finding a specific function or θ where the constructed products on the quantum torus violate the conditions required for the non-inequality, or computing an explicit counterexample to the non-inequality in the quantum case.

read the original abstract

We demonstrate a construction of products on the quantum torus $\mathbb{T}_\theta^2$ that generalises the usual construction of finite Riesz products on the commutative torus $\mathbb{T}^2$. We explain why the former constitutes a natural analogue of the latter in the non-commutative setting and, based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs finite Riesz products on the quantum torus T_θ² that generalize the classical finite Riesz products on the commutative torus T². It argues that the construction is a natural non-commutative analogue and, combining it with prior theorems of Kazaniecki and the second author, proves a non-commutative version of Ornstein's non-inequality.

Significance. If the construction is rigorously shown to be a natural analogue and the cited prior results apply without additional hypotheses failing under the cocycle deformation, the work would extend a classical result in harmonic analysis to the setting of quantum tori. This could be of interest to researchers in operator algebras and non-commutative ergodic theory. No machine-checked proofs or reproducible code are mentioned, but the explicit generalization and reliance on established results constitute a clear strength if the applicability is verified.

major comments (1)
  1. [construction and application of prior results (around the invocation of Kazaniecki–Wojciechowski theorems)] The central proof invokes prior results of Kazaniecki and the second author to obtain the Ornstein non-inequality from the new construction. However, those results were established under hypotheses (L¹-boundedness, weak-* convergence, spectral properties) that hold automatically in the commutative case but may require separate verification once multiplication is deformed by the cocycle θ. The manuscript should contain an explicit check (e.g., in the section presenting the construction and its properties) that all such hypotheses remain valid for θ ≠ 0; without it the non-inequality does not follow directly from the cited theorems.
minor comments (2)
  1. [introduction and motivation for the construction] Clarify the precise sense in which the quantum-torus products are claimed to be 'natural analogues'—e.g., which algebraic or analytic properties are preserved and which are deformed.
  2. [preliminaries] Ensure consistent notation for the quantum torus T_θ² and the associated von Neumann algebra throughout; avoid switching between different presentations of the cocycle without explicit cross-reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that merits clarification. We address the major comment below and will revise the paper accordingly to strengthen the argument.

read point-by-point responses

  1. Referee: [construction and application of prior results (around the invocation of Kazaniecki–Wojciechowski theorems)] The central proof invokes prior results of Kazaniecki and the second author to obtain the Ornstein non-inequality from the new construction. However, those results were established under hypotheses (L¹-boundedness, weak-* convergence, spectral properties) that hold automatically in the commutative case but may require separate verification once multiplication is deformed by the cocycle θ. The manuscript should contain an explicit check (e.g., in the section presenting the construction and its properties) that all such hypotheses remain valid for θ ≠ 0; without it the non-inequality does not follow directly from the cited theorems.

    Authors: We agree that an explicit verification of the hypotheses from Kazaniecki–Wojciechowski is necessary to ensure the prior theorems apply directly under the cocycle deformation. While the construction in Section 3 is designed so that the finite Riesz products retain L¹-boundedness (via the same trigonometric polynomial estimates, which are independent of θ), weak-* convergence (by continuity of the deformed multiplication in the weak-* topology), and the relevant spectral properties (as the support of the spectrum is preserved under the cocycle action), these facts are only implicit in the current text. We will add a dedicated paragraph or short subsection immediately following the construction (in the section presenting the finite Riesz products on T_θ²) that verifies each hypothesis holds uniformly for all θ. This will make the invocation of the prior results fully rigorous without additional assumptions. revision: yes

Circularity Check

1 steps flagged

Central non-inequality obtained via direct invocation of prior results by the second author without separate hypothesis verification for the quantum-torus construction

specific steps
  1. self citation load bearing [Abstract]
    "based on this construction, as well as on previous results by K. Kazaniecki and the second author, we prove a non-commutative version of an Ornstein non-inequality."

    The load-bearing step that yields the paper's main theorem reduces to invoking prior results whose author list overlaps with the present paper. No independent verification is supplied that the hypotheses of those results continue to hold once the torus multiplication is deformed by the non-commutative cocycle θ; the non-inequality is therefore obtained by direct application of the self-cited theorems rather than by a fresh derivation from the new construction alone.

full rationale

The paper introduces an original construction of finite Riesz products on the quantum torus T_θ² that generalizes the commutative case and argues it is a natural analogue. This part of the derivation is self-contained. However, the proof of the main result (the non-commutative Ornstein non-inequality) is explicitly based on applying previous theorems by K. Kazaniecki and the second author. Because those theorems were established in the commutative setting, their direct applicability requires checking that all hypotheses (e.g., L¹-boundedness or weak-* convergence) survive the cocycle deformation; the paper treats the application as immediate. This constitutes moderate self-citation load-bearing but does not render the entire derivation circular by construction, as the new construction supplies independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted; the work appears to rest on standard background from operator algebras and the referenced prior results.

pith-pipeline@v0.9.0 · 5379 in / 1216 out tokens · 47375 ms · 2026-05-09T22:29:39.109120+00:00 · methodology

discussion (0)

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Reference graph

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