On the nonarchimedean quadratic Lagrange spectra

TL;DR

This paper explores Diophantine approximation in nonarchimedean fields, introducing a Lagrange spectrum for quadratic irrationals and establishing analogs of classical real case results using geometric group actions.

Contribution

It defines a nonarchimedean Lagrange spectrum for quadratic irrationals and proves key properties like closedness, boundedness, and the existence of a Hall ray, using Bruhat-Tits trees.

Findings

Lagrange spectrum is closed and bounded in nonarchimedean fields.

Existence of a Hall ray in the spectrum.

Computed Hurwitz constants for these fields.

Abstract

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.