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arxiv: 2601.10360 · v1 · submitted 2026-01-15 · 🧮 math.CA

On UC-multipliers for multiple trigonometric systems

Grigori A. Karagulyan This is my paper

Pith reviewed 2026-05-16 14:15 UTC · model grok-4.3

classification 🧮 math.CA
keywords Weyl multiplierstrigonometric systemsalmost everywhere convergencerearrangementsequivalence principleFourier seriesmultidimensional analysis
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The pith

Any sequence serving as a Weyl multiplier for all rearrangements of multiple trigonometric systems must satisfy log n ≲ w(n) ≲ log² n.

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

The paper investigates sequences w(n) that serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. It proves that any such sequence must obey the bounds log n less than or equal to w(n) less than or equal to log squared n. This follows from establishing a general equivalence principle that reduces multidimensional problems directly to the one-dimensional case. A sympathetic reader cares because the result pins down the precise growth rate separating convergent and divergent Fourier series under arbitrary rearrangements in higher dimensions.

Core claim

The central claim is that the class of sequences w(n) serving as a.e. convergence Weyl multipliers for all rearrangements of multiple trigonometric systems is characterized by the bounds log n ≲ w(n) ≲ log² n. This characterization is obtained by proving a general equivalence principle between one-dimensional and multidimensional trigonometric systems that allows direct transfer of one-dimensional estimates to the higher-dimensional setting without additional restrictions.

What carries the argument

The equivalence principle between one-dimensional and multidimensional trigonometric systems, which permits direct transfer of multiplier estimates from the simpler case to the multi-dimensional case.

If this is right

  • One-dimensional multiplier estimates extend immediately to the multi-dimensional case under the same conditions.
  • The lower bound log n ≲ w(n) is necessary for convergence even after arbitrary rearrangements.
  • The upper bound w(n) ≲ log² n is sufficient for a.e. convergence in the multi-dimensional setting.
  • Known one-dimensional results on Weyl multipliers apply verbatim to higher-dimensional trigonometric systems.

Where Pith is reading between the lines

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

  • The same equivalence might allow reduction of other convergence questions in harmonic analysis to the one-dimensional setting.
  • Similar principles could classify multipliers for rearranged systems built from other orthogonal bases.
  • The bounds suggest that numerical tests on finite partial sums of rearranged multi-dimensional series could verify the transition between convergence and divergence.

Load-bearing premise

The equivalence principle between one- and multi-dimensional systems holds without extra restrictions on rearrangements or measure spaces.

What would settle it

A concrete counterexample would be a sequence w(n) satisfying log n ≲ w(n) ≲ log² n together with a specific rearrangement of a multi-dimensional trigonometric system whose Fourier series diverges on a set of positive measure.

read the original abstract

We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions.

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 / 1 minor

Summary. The manuscript investigates the class of sequences w(n) that serve as almost-everywhere convergence Weyl multipliers (UC-multipliers) for all rearrangements of multiple trigonometric systems. It proves that any such sequence must satisfy the bounds log n ≲ w(n) ≲ log² n. The central result is a general equivalence principle between one-dimensional and multidimensional trigonometric systems that permits direct transfer of one-dimensional estimates to the higher-dimensional setting.

Significance. If the equivalence principle holds in full generality for arbitrary measure spaces and rearrangements, the bounds would be sharp and would constitute a meaningful extension of known one-dimensional multiplier results to the multi-dimensional case. The paper supplies a transfer mechanism that, if verified without hidden restrictions, would be a useful technical tool for the field.

major comments (1)
  1. [Main theorem on equivalence principle] The equivalence principle (main result): the statement claims generality sufficient to transfer one-dimensional bounds directly, but the manuscript must explicitly confirm that the principle applies to arbitrary (non-product) measure spaces and rearrangements that do not preserve coordinate independence. This verification is load-bearing for the claimed bounds log n ≲ w(n) ≲ log² n on all multiple systems; without it the upper and lower estimates may hold only under additional structural assumptions.
minor comments (1)
  1. [Abstract] The abstract introduces 'UC-multipliers' without a one-sentence definition or forward reference; adding this would improve immediate readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the scope of the equivalence principle. We address this point directly below and will incorporate a clarifying revision.

read point-by-point responses

  1. Referee: The equivalence principle (main result): the statement claims generality sufficient to transfer one-dimensional bounds directly, but the manuscript must explicitly confirm that the principle applies to arbitrary (non-product) measure spaces and rearrangements that do not preserve coordinate independence. This verification is load-bearing for the claimed bounds log n ≲ w(n) ≲ log² n on all multiple systems; without it the upper and lower estimates may hold only under additional structural assumptions.

    Authors: The equivalence principle is formulated and proved specifically for multiple trigonometric systems on the d-dimensional torus with its standard product Lebesgue measure. The argument proceeds by reducing the multi-dimensional rearranged system to a one-dimensional system via a suitable identification of indices and does not impose any requirement that the rearrangement preserve coordinate independence; it applies to an arbitrary bijection of the multi-index set. We do not claim the result for completely arbitrary non-product measure spaces, as the trigonometric system itself is defined via the product structure of the torus. Within the setting of the paper, the equivalence therefore permits direct transfer of the known one-dimensional bounds, yielding log n ≲ w(n) ≲ log² n for all rearrangements of the multiple system. To meet the referee's request for explicit confirmation, we will add a short clarifying paragraph immediately after the statement of the main theorem (and a corresponding sentence in the introduction) that records the precise measure-space assumptions and notes that the proof accommodates arbitrary rearrangements. This revision will be limited to exposition and will not change the scope or the proofs. revision: partial

Circularity Check

0 steps flagged

No circularity: equivalence principle is independent transfer from 1D results

full rationale

The paper establishes a general equivalence principle as its main theorem, which transfers known one-dimensional estimates to the multidimensional setting without any self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain visible in the abstract or derivation outline. The bounds log n ≲ w(n) ≲ log² n are obtained by applying this independently proven equivalence to prior 1D results on UC-multipliers, rather than deriving them by construction from the multiD case itself. No equations or steps reduce the central claim to its own inputs or to unverified self-citations; the structure remains self-contained against external 1D benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; the work rests on standard real-analysis background for trigonometric series and rearrangements.

pith-pipeline@v0.9.0 · 5351 in / 1024 out tokens · 54315 ms · 2026-05-16T14:15:46.590280+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Quantitative estimates for the absolute convergence of wavelet-type series

    math.CA 2026-04 unverdicted novelty 6.0

    The sum 1/(n w(n)) converging is necessary and sufficient for an increasing w(n) to be an a.e. unconditional convergence Weyl multiplier for arbitrary wavelet-type systems, and log n is optimal for rearranged systems.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

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