Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields

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.