Lagrange spectrum of a circle over the Eisensteinian field

TL;DR

This paper investigates the Lagrange spectrum of the unit circle over the Eisensteinian field, establishing its minimum, smallest accumulation point, and characterizing the spectrum's values in a specific range.

Contribution

It provides the first detailed analysis of the Lagrange spectrum for the circle over the Eisensteinian field, including key spectral values and structure.

Findings

Minimum of the spectrum is 2

Smallest accumulation point is 4/√3

Characterization of spectrum values between 2 and 4/√3

Abstract

We study an intrinsic Lagrange spectrum of the unit circle $|z|=1$ in the complex plane with respect to the Eisensteinian field $\mathbb{Q}(\sqrt{-3})$. We prove that the minimum of the Lagrange spectrum is $2$ and that its smallest accumulation point is $4/\sqrt{3}$. In addition, we characterize the set of all values in the spectrum between $2$ and $4/\sqrt3$.