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Diophantine Approximation in local function fields via Bruhat-Tit trees

Luis Arenas-Carmona, Claudio Bravo

Abstract

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.

Diophantine Approximation in local function fields via Bruhat-Tit trees

Abstract

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.
Paper Structure (12 sections, 23 theorems, 37 equations, 6 figures)

This paper contains 12 sections, 23 theorems, 37 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Conventions on graphs
  3. The Bruhat-Tits tree of $\mathrm{SL}_2$
  4. Arithmetic Quotients
  5. The walk of a local integer
  6. Approximation and quotient graphs
  7. Proof of Theorem \ref{['T3']}.
  8. Proof of Theorem \ref{['T2']}.
  9. Random processes and Haar measure
  10. Proof of Theorem \ref{['T1']}
  11. Proof of Theorem \ref{['T1b']}
  12. Examples

Key Result

Theorem 1.1

The set $\Psi_M\cap\mathcal{O}_\infty$ has full Haar measure for every integer $M$. Furthermore, there exists a set $\Sigma$ of full Haar measure such that every element $\alpha\in\Sigma$ is the limit of a sequence $\left\{\frac{f_n}{g_n}\right\}_n$ such that

Figures (6)

  • Figure 1: Figure (A) shows the $\mathbf{S}$-graph when $\mathbb{P}^1_{\mathbb{F}}$ and $\deg(P_{\infty})=2$. Figure (B) is the corresponding classifying graph, which has a semi-edge joined to $\mathbf{w}_0$.
  • Figure 2: Figures used in the proof of Lemma \ref{['PWA2']}
  • Figure 3: Diagrams used in the proofs of Lemmas \ref{['L52']} and Lemma \ref{['L53']}. The box marked with a "$Z$" denotes a connected subgraph.
  • Figure 4: Finding explicit elements with a prescribed path.
  • Figure 5: The $\mathbf{S}$-graph $\Gamma \backslash \mathfrak{t}$ when $\mathbb{P}^1_{\mathbb{F}}$ and $\deg(P_{\infty})=3$. The edge represented by a wiggly line equals the central graph of $\Gamma \backslash \mathfrak{t}$.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 39 more