Intrinsic Diophantine approximation on circles and spheres
arxiv(2024)
摘要
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.
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