On the nonarchimedean quadratic Lagrange spectra

Jouni Parkkonen , Frédéric Paulin (1)

1 Topologie, géométrie, dynamique

We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.

Mots clés: quadratic irrational; continued fraction expansion; positive characteristic; formal Laurent series; Lagrange spectrum; Hurwitz constant; Hall ray

Domaines: Théorie des nombres [math.NT]; Théorie des groupes [math.GR]

Jouni Parkkonen, Frédéric Paulin. On the nonarchimedean quadratic Lagrange spectra. 2018. ⟨hal-01697088⟩