有限體下正規Laurent 級數上的Duffin-Schaeffer 猜想

Abstract

我們將在本論文中探討有限體下正規Laurent級數上的Diophantine逼近。更進一步地，在此深入研究Duffin-Schaeffer猜想，並證明一些特殊情況下的結果。本文的主要架構如下：在第一章，我們將介紹Diophantine逼近及賦距數論，並藉由實數中一些已知的結果，作為我們研究動機。

在第二章，我們將介紹在正規Laurent級數體上的Diophantine逼近。第一節羅列了一些定義及我們即將使用的符號。在這一節，我們發展一些重要的基本性質，例如：廣義Borel-Cantelli引理及數論中類似的結果等。第二節是近來有關這個主題研究結果的整理。在這節裡，我們探討各種情況下的Diophantine逼近並給出相對應的結果。除此之外，在第二節也明確指出Duffin-Schaeffer猜想的目標。第三節，則是我們對這個猜想所貢獻的結果。

在三章，我們給出第二章第三節所談主要結果的詳細證明。簡短地說，我們先估計任兩事件交集發生的機率大小，接著應用廣義的Borel-Cantelli引理及零一律。主要證明的架構，模仿實數情況中Vaaler的證法。

在第四章，我們改變探討猜想的觀點，與實數情況中Harman的部分結果相似。

在第五章，論文的尾聲，我們將針對整體工作做一個總結。

This thesis is concerned with metric Diophantine approximation for formal Laurent series over a finite base field. More precisely, we will discuss an analogue of the famous Duffin-Schaeffer conjecture for formal Laurent series and prove it in some special cases.

An outline of the thesis is as follows. In Chapter 1, we will briefly introduce Diophantine approximation and metric Diophantine approximation over the real number field and state some results which are important for our work. In Chapter 2, we will give an introduction into the theory of Diophantine approximation for formal Laurent series over a finite base field. More precisely, Section 2.1 will collect the definitions, notations and results we are going to use throughout this work. Then, in Section 2.2 we will give a survey on recent research activities in Diophantine approximation for formal Laurent series. Apart from such results, this section will also be used to state the Duffin-Schaeffer conjecture in our context and explain the goal of this thesis in more details. Finally, Section 2.3 will contain our findings concerning this conjecture.

In Chapter 3, we will give details of the proof of our main result. Roughly speaking, we will follow the classical path which is concerned with estimating the measure of the intersection of two events, and applying the generalized Borel-Cantelli lemma.

In Chapter 4, we will consider the conjecture from a different angle and prove some analogous results of Harman.

Finally, we will end the thesis with a conclusion in Chapter 5.