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 R2$\mathbb {R}<^>2$ or R3$\mathbb {R}<^>3$ and three spheres embedded in R3$\mathbb {R}<^>3$ or R4$\mathbb {R}<^>4$. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of R$\mathbb {R}$ and C$\mathbb {C}$. Thanks to prior work of Asmus L. Schmidt on the spectra of R$\mathbb {R}$ and C$\mathbb {C}$, 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.