Intrinsic Diophantine approximation on circles and spheres

Abstract

We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ and three spheres embedded in ${\mathbb{R}}^3$ or ${\mathbb{R}}^4$. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of $\mathbb{R}$ and $\mathbb{C}$. Thanks to prior work of Asmus L.~Schmidt on the spectra of $\mathbb{R}$ and $\mathbb{C}$, we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.

28 pages, 9 figures