Lagrange spectrum of a circle over the Eisensteinian field

Byungchul Cha; Heather Chapman; Brittany Gelb; Chooka Weiss

Lagrange spectrum of a circle over the Eisensteinian field

Number Theory 2021-10-29 v2

Authors: Byungchul Cha , Heather Chapman , Brittany Gelb , Chooka Weiss

Abstract

We study an intrinsic Lagrange spectrum of the unit circle $|z|=1$ in the complex plane with respect to the Eisensteinian field $\mathbb{Q}(\sqrt{-3})$ . We prove that the minimum of the Lagrange spectrum is $2$ and that its smallest accumulation point is $4/\sqrt{3}$ . In addition, we characterize the set of all values in the spectrum between $2$ and $4/\sqrt3$ .

Keywords

lagrange spectrum eisenstein field unit circle accumulation point diophantine approximation

Cite

@article{arxiv.2003.13188,
  title  = {Lagrange spectrum of a circle over the Eisensteinian field},
  author = {Byungchul Cha and Heather Chapman and Brittany Gelb and Chooka Weiss},
  journal= {arXiv preprint arXiv:2003.13188},
  year   = {2021}
}

Comments

44 pages, 10 figures, to be published in Monatsh. Math

Lagrange spectrum of a circle over the Eisensteinian field

Abstract

Keywords

Cite

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