Diophantine Approximation in local function fields via Bruhat-Tit trees

Luis Arenas-Carmona, Claudio Bravo

Published 2024-01-10Version 1

We use the theory of arithmetic quotients of the Bruhat-Tits tree developed by Serre and others to obtain Dirichlet-style theorems for Diophantine approximation on global function fields. This approach allows us to find sharp values for the constants involved and, occasionally, explicit examples of badly approximable quadratic irrationals. Additionally, we can use this method to easily compute the measure of the set of elements that can be written as the limit of a sequence of ``better than expected'' approximants. All these results can be easily obtained via continued fractions when they are available, so that quotient graphs can be seen as a partial replacement of them when this fails to be the case.