Intrinsic Diophantine approximation on circles and spheres

TL;DR

This paper investigates the intrinsic Diophantine approximation spectra of certain circles and spheres, establishing connections with spectra of real and complex numbers, and determining key spectral features for these geometric objects.

Contribution

It introduces a unified framework linking spectra of circles and spheres with those of nd alculus, and explicitly characterizes their smallest accumulation points and initial discrete parts.

Findings

Identifies the smallest accumulation points of the spectra.

Determines the initial discrete parts of the spectra.

Establishes connections with spectra of nd alculus.

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.