Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

TL;DR

This paper explores the nonarchimedean quadratic Lagrange spectrum in power series fields, introducing a new approach based on continued fractions to analyze approximation properties, complementing previous geometric methods.

Contribution

It presents a novel method using continued fractions in power series fields to study the quadratic Lagrange spectrum, expanding beyond geometric group action techniques.

Findings

Developed a continued fractions framework for power series fields

Connected continued fractions with quadratic approximation spectra

Provided new insights into nonarchimedean approximation dynamics

Abstract

Let ${{\bf F}}_q$ be a finite field of order a positive power $q$ of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given quadratic power series in ${{\bf F}}_q((Y^{-1}))$, for the action by homographies and anti-homographies of ${\rm PGL}_2({{\bf F}}_q[Y])$ on ${{\bf F}}_q((Y^{-1})) \cup \{\infty\}$. While their approach used geometric methods of group actions on Bruhat--Tits trees, ours is based on the theory of continued fractions in power series fields.