On finiteness of spectral radius order

TL;DR

This paper characterizes numbers with finite spectral radius order within quadratic algebraic integers and numbers up to 2, providing explicit values for specific families, which advances understanding of spectral radius order's properties.

Contribution

It offers a characterization of finite spectral radius order for quadratic algebraic integers and numbers no larger than 2, including explicit calculations for certain families.

Findings

Characterization of finite spectral radius order in quadratic algebraic integers.

Explicit spectral radius order values for two infinite families.

Extension of spectral radius order understanding to specific classes.

Abstract

The concept of spectral radius order plays an crucial role in the breakthrough work on equiangular lines due to Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. (2) 194 (2021), no. 3, 729-743]. However, it is difficult to calculate the spectral radius order explicitly in general, or even to characterize numbers with finite spectral radius order. In this paper, we characterize numbers with finite spectral radius orders in two special classes: quadratic algebraic integers and the numbers no larger than 2. Additionally, we derive precise values of the spectral radius order of two infinite families of quadratic algebraic integers.