$\lambda$-quiddit{\'e} sur certains sous-groupes monog{\`e}nes de $\mathbb{C}$

TL;DR

This paper investigates the properties of $\lambda$-quiddities within certain cyclic subgroups of the complex numbers, focusing on those generated by square roots and imaginary multiples of natural numbers, extending the concept's application.

Contribution

It extends the study of $\lambda$-quiddities to specific cyclic subgroups of $\mathbb{C}$, particularly those generated by $\sqrt{k}$ and $i\sqrt{k}$, which was not previously explored.

Findings

Characterization of $\lambda$-quiddities in cyclic subgroups generated by $\sqrt{k}$.

Analysis of $\lambda$-quiddities in subgroups generated by $i\sqrt{k}$.

Extension of Coxeter's frieze concepts to new subsets of $\mathbb{C}$.

Abstract

During the study of Coxeter's friezes, M. Cuntz defined the concept of $\lambda$-quiddities and gave the problem of studying them over some subsets of $\mathbb{C}$. The objective of this text is to carry out this study in the case of some cyclic subgroups of ($\mathbb{C},+$). In particular we will study the case of the cyclic subgroups generated by $\sqrt{k}$ and $i\sqrt{k}$, with $k \in \mathbb{N}$.