Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 31 min ago
PreprintCC BY 4.0arXiv

Existence of a Hall ray in higher dimensional Lagrange spectra

  • Yitwah Cheung
  • Zhijing Wendy Wang
  • Ruichong Zhang
Open at sourcePDFDOICite

Abstract

Given any norm |·|ℝ2 on ℝ2, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small ρ≥0 occurs as $\liminf_{q\to\infty}q^{1/2}\min_{p\in\mathbb Z^2}\|qx-p\|_{\mathbb R^2} $ for some x∈ℝ2∖ℚ2. Combined with a recent result of Kleinbock \cite{Kleinbock}, this implies that the $2$-dimensional Lagrange spectrum is either a ray, mirroring the behavior for Dirichlet spectra, or a ray preceded by a dense part.

Related

  1. An Introduction to the Lagrange and Markov Spectra through the Lens of Generalized Markov Numbers
    Yasuaki Gyoda
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Distribution of points near the origin in the d-dimensional Lagrange spectrum
    Benjamin Ward
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryDynamical systemsarXiv
  3. Existence of Weak Solutions and Higher-Order Regularity for a Heat-Wave Fluid-Structure Interaction System on a Periodic Strip
    Mohammad Mahabubur Rahman
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Normed idempotence of spectra and distributivity in equivariant higher algebra
    Kaif Hilman, Branko Juran, Natalie Stewart
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  5. Existence of multi-dimensional pulsating fronts for KPP equations: a new formulation approach
    Liangliang Deng, Arnaud Ducrot
    Calculus of Variations and Partial Differential EquationsPeer-reviewedCC BY 4.0Full text5 citedGeneral mathematicsHAL

Cite

BibTeXRIS
BibTeX
@misc{cheung2026existence,
  title = {{Existence of a Hall ray in higher dimensional Lagrange spectra}},
  author = {Yitwah Cheung and Zhijing Wendy Wang and Ruichong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23464},
  eprint = {2609.23464},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23464},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Existence of a Hall ray in higher dimensional Lagrange spectra
AU  - Yitwah Cheung
AU  - Zhijing Wendy Wang
AU  - Ruichong Zhang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23464
UR  - https://arxiv.org/abs/2609.23464
AB  - Given any norm |·|_ℝ^2 on ℝ^2, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small ρ≥0 occurs as $\liminf_{q\to\infty}q^{1/2}\min_{p\in\mathbb Z^2}\|qx-p\|_{\mathbb R^2} $ for some x∈ℝ^2∖ℚ^2. Combined with a recent result of Kleinbock \cite{Kleinbock}, this implies that the $2$-dimensional Lagrange spectrum is either a ray, mirroring the behavior for Dirichlet spectra, or a ray preceded by a dense part.
ER  -