arXiv:2607.25610v1
Brjuno condition through best approximations and the linearization problem
math.DS
Abstract
We consider the classical analytic linearization problem for vector fields on the torus close to a constant vector field . Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with under a diagonal flow on . Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.
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Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
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Detailed mathematical audit
01Statements3 reported findingsCorrect
The equivalent Brjuno conditions for linear forms and the quantitative analytic linearization theorem are correct. The displayed notation slips identified below do not alter any mathematical claim.
Theorem 1.1Correct
The geometric, best-approximation, and integral Brjuno criteria agree
Pages 3 and 6–14 · Theorem 1.1 and Section 3 · arXiv:2607.25610v1
The switching times of the diagonal-flow shortest-vector function correspond to successive best approximations. On each affine segment the integral contribution is comparable to the associated logarithmic best-approximation term, and the endpoint estimates make all four summability conditions equivalent.
Full paper, version 1 ↗Theorem 1.2Correct
The announced analytic conjugacy follows with the explicit Brjuno loss
Pages 4 and 14–21 · Theorem 1.2 and Section 4 · arXiv:2607.25610v1
The far-from-resonance homological equation is solved with the stated divisor bound, the retained resonance cone is narrowed at every switching time, and the accumulated analyticity loss is exactly the convergent Brjuno quantity controlled in Theorem 1.1.
Local displayed notationTypos · no status impact
The displayed symbol and index mismatches are typos
Pages 11, 18, 30, and 33–34 · Lemma 3.13, Proposition 4.8, and Appendices B–C · arXiv:2607.25610v1
The intermediate inequality in Lemma 3.13 needs , not the reversed difference. Proposition 4.8 needs in its condition, not the unbound . Appendix B must read and Lemma B.3 needs , without the stray subscript . In Appendix C, the norm-equivalence chain must begin with , and the last ratio must use in its denominator. Every repair is fixed by the immediately adjacent definition or equality and leaves the estimates unchanged.
02Proofs1 reported findingCorrect
The arithmetic comparison and analytic iteration are complete after the local typo corrections; all convergence estimates use compatible constants and quantifiers.
Sections 3–4 and Appendices A–CCorrect and complete after the stated repairs
The proof chains close after the stated symbol repairs
Pages 6–35 · arithmetic equivalences, iterative scheme, and appendices · arXiv:2607.25610v1
The best-approximation intervals partition the relevant flow scales, while the Banach-space estimates control composition, inversion, and the constant term. The chosen initial radius dominates the sum of later losses, so the coordinate changes converge to the claimed analytic conjugacy.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.