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<p>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 particular sets of elements that can be approximated better than predicted by these theorems. 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.</p>

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Diophantine Approximation in local function fields via Bruhat-Tits trees

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Luis Arenas-Carmona,
  • Claudio Bravo

摘要

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 particular sets of elements that can be approximated better than predicted by these theorems. 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.