Intrinsic Diophantine approximation on circles and spheres

초록

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.