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Abstract

Let $\mathbb F_q$ 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 $\mathbb F_q((Y^{-1}))$, for the action by homographies and anti-homographies of ${\rm PGL}_2(\mathbb F_q[Y])$ on $\mathbb F_q((Y^{-1})) \cup \{\infty \}$. While Parkkonen and Paulin’s approach used geometric methods of group actions on Bruhat–Tits trees, ours is based on the theory of continued fractions in power series fields.