Skip to content
DraconiticResearch
Search Sources API
190,335 studies31 sourcesUpdated 16 min ago
PreprintarXiv licencearXiv

Non-degeneracy and Fibering with Applications to Diophantine Approximation

  • ZiAn Cody Zhao
Open at sourcePDFDOICite

Abstract

Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in ℝn, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be CN with the essentially optimal bound N≥ max{ n,l+1}, where the manifold is l non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small CN perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.

Related

  1. Metric Diophantine approximation on planar and Veronese curves
    Mumtaz Hussain, Johannes Schleischitz, David Simmons
    International Journal of Number TheoryPeer-reviewedLicence not statedFull text5 citedAlgebra & number theoryOpenAlex
  2. Extreme value laws for intrinsic Diophantine approximation on spheres
    Alexander Gorodnik, Zouhair Ouaggag
    arXivPreprintarXiv licenceFull textAlgebra & number theoryDynamical systemsarXiv
  3. Diophantine approximation by primes and Landau--Siegel zeros
    Sun-Kai Leung, Stelios Sachpazis
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. On a Notion of Density Related to Restricted Diophantine Approximation
    Chance Sanford
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Twisted Diophantine Approximation I: Asymptotic Theory
    Taehyeong Kim, Vasiliy Neckrasov
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{zhao2026nondegeneracy,
  title = {{Non-degeneracy and Fibering with Applications to Diophantine Approximation}},
  author = {ZiAn Cody Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10799},
  eprint = {2610.10799},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10799},
}
RIS
TY  - UNPB
TI  - Non-degeneracy and Fibering with Applications to Diophantine Approximation
AU  - ZiAn Cody Zhao
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10799
UR  - https://arxiv.org/abs/2610.10799
AB  - Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in ℝ^n, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be C^N with the essentially optimal bound N≥ max{ n,l+1}, where the manifold is l non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small C^N perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.
ER  -