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arxiv: 2410.04583 · v2 · submitted 2024-10-06 · 🧮 math.CV · math.DS

A new quasi-analytic class

Mauricio Garay , Duco van Straten This is my paper

Pith reviewed 2026-05-23 20:04 UTC · model grok-4.3

classification 🧮 math.CV math.DS
keywords quasi-analytic classesKAM theoryperturbation theorydivergent seriespole accumulationdynamical systemscomplex analysisTaylor expansions
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The pith

A new quasi-analytic class accounts for divergences in KAM theory expansions via pole accumulations.

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

The paper defines spaces of quasi-analytic classes where Taylor expansions exist and are unique but may not converge. Unlike Borel or Denjoy-Carleman examples, this new class fits expansions common in perturbation theory, KAM theory, and dynamical systems. It provides a geometric explanation for why most such series diverge, namely through the presence of accumulation points of poles. This matters for understanding the limitations and behavior of perturbative methods in mathematical physics.

Core claim

We introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.

What carries the argument

The newly introduced quasi-analytic class, defined by existence and uniqueness of Taylor expansions on domains with controlled pole accumulations.

If this is right

  • The class applies directly to divergent perturbative expansions in dynamical systems.
  • Divergence is explained by accumulation points of poles in the complex plane.
  • Most perturbative expansions from physics and number theory fall into this class rather than previous ones.
  • Quasi-analyticity holds despite non-convergence due to the geometric setup.

Where Pith is reading between the lines

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

  • Similar classes might be constructed for other contexts with divergent series, like quantum mechanics.
  • This geometric view could inspire new resummation techniques for KAM expansions.
  • Applications may extend to number theory problems involving divergent series.

Load-bearing premise

The new class satisfies the quasi-analytic property and its construction captures the divergence mechanism from pole accumulations in KAM theory.

What would settle it

An example of a KAM perturbative expansion that diverges without accumulation points of poles, or converges despite such points.

read the original abstract

Spaces of quasi-analytic classes are defined by the existence and uniqueness of Taylor expansions, which are not necessarily convergent. First examples were given by Borel in his theory of monogenic functions, a generalisation of holomorphic functions defined on locally closed sets. Denjoy and Carleman then gave simpler examples of quasi-analytic classes which are now widely known. Unfortunately, in most examples coming from mathematical physics and number theory, the power series are neither of Borel nor Denjoy-Carleman's classes. In this paper we introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.

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 / 0 minor

Summary. The manuscript introduces a new quasi-analytic class of functions, distinct from the Borel and Denjoy-Carleman classes, defined by the existence and uniqueness of (possibly divergent) Taylor expansions. It claims this class is relevant to perturbation theory, KAM theory, and dynamical systems, with theorems that geometrically explain the divergence of most perturbative expansions via the presence of accumulation points of poles.

Significance. If the construction and theorems were substantiated, the work could offer a new framework for analyzing divergent perturbative series in mathematical physics and dynamical systems. However, the provided manuscript contains only the abstract with no definitions, theorem statements, or proofs, so no assessment of significance is possible.

major comments (1)
  1. [Abstract] The manuscript consists solely of the abstract; no definition of the proposed quasi-analytic class is given, nor are any theorems stated or proved. This is load-bearing for the central claim, as the quasi-analytic property and the geometric explanation of divergence cannot be verified without the construction and arguments.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The observation that the submitted manuscript contains only the abstract is correct, and we will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] The manuscript consists solely of the abstract; no definition of the proposed quasi-analytic class is given, nor are any theorems stated or proved. This is load-bearing for the central claim, as the quasi-analytic property and the geometric explanation of divergence cannot be verified without the construction and arguments.

    Authors: We agree with this assessment of the submitted file. The revised version will contain the explicit definition of the new quasi-analytic class (via existence and uniqueness of possibly divergent Taylor expansions on suitable domains), the precise statements of the main theorems, and their proofs, including the geometric argument linking divergence to accumulation points of poles. This will permit direct verification of the quasi-analyticity and the claimed relevance to perturbation and KAM theory. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a new quasi-analytic class via an explicit construction tied to perturbation theory and KAM contexts, then states theorems establishing the quasi-analytic property (existence and uniqueness of Taylor series) and the geometric divergence mechanism via pole accumulation. No load-bearing step reduces by definition, by fitted parameter, or by self-citation chain to its own inputs; the central claims rest on independent verification of the stated properties rather than tautological renaming or ansatz smuggling. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities are stated.

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Reference graph

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22 extracted references · 22 canonical work pages · 1 internal anchor

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