在正規Laurent 級數體探討關於Kurzweil 定理之延伸結果

Abstract

在過去十年裡，已經有了許多關於正規Laurent 級數體下之賦距Diophantine逼近的研究，而最近這項研究有了一個有趣的新研究方向──關於Kurzweil 定理的改良。本論文主要的工作就是總結整理這些不同方向的改良以及提供一些新的貢獻。 其中一項改良是由Kim、Nakada 和Natsui 在[6]中所提出，在本文中，我們將指出他們提出的證明當中，有部分細節是可以被改進的;更精確來說，我們可 以將其中的單調性條件拿掉，而這可以讓我們重新證明Kurzweil 定理的其中一 個方向。本文另一個主題是關於Kurzweil 本身在[8]中所提出的在實數體上的另 一個改良，我們將證明這個定理在正規Laurent 級數體下的相似結果並與另一個 最近由Kim、Tan、Wang 與Xu 在[7]提出的與之相似的改良作比較。 本文的主要架構如下：我們將在第一章介紹一些關於Diophantine 逼近的背 景知識以及本論文的目標。第一節中，我們將回顧有關Diophantine 逼近的基本 性質和介紹我們所用的符號。在第二節，我們將介紹Diophantine 逼近和賦距 Diophantine 逼近，這可以分成homogeneous 和inhomogeneous 的情形。我們收集一些關於這兩個情形的結果，尤其是包含所謂的double-metric 和single-metric 兩種情況的inhomogeneous 情形。最後，我們將在第三節介紹本論文的主要目的。 在第二章，我們將探討Kim、Nakada 和Natsui 所提出的改良。我們將在第 一節陳述一些引理和呈現他們在改良中的證明。第二節中，我們將改進前一節中 的證明，並利用改進後的結果來證明Kurzweil 定理的其中一個方向。第三節， 我們將利用完全不同於第一節的證明方法來證明一個特殊情形。 我們在第三章證明一個與Kurzweil 在[8]提出之改良相似的結果。因為這個 結果與另一個最近由Kim、Tan、Wang 與Xu 提出的改良有些異同處，我們將在 這個章節的最後讓兩者作些比較。 最後，在第四章，我們將提出一些猜想來對本論文做一個總結。

The last decade has witnessed a lot of research about metric Diophantine approximation in the field of formal Laurent series, where a recent new and interesting research direction was concerned with refinements of Kurzweil’s theorem. The purpose of this thesis is to summarize these refinements and give some new contributions. One of these refinements was given by Kim, Nakada and Natsui in [6]. In this thesis, we will show that some details of their proofs can be improved. More precisely, we are able to drop the monotonicity condition and this will allow us to reprove one direction of Kurzweil’s theorem. Another topic of this thesis will be concerned with another refinement of Kurzweil’s theorem which in the real case was obtained by Kurzweil himself in [8]. We will prove an analogue of this theorem in the field of formal Laurent series and compare it with another refinement of a similar flavour which was recently proved by Kim, Tan, Wang and Xu in [7]. An outline of this thesis is as follows. In Chapter 1, we will introduce some background knowledge on Diophantine approximation and explain our aim of this thesis. There are three sections in this chapter. In Section 1.1, we will recall some fundamental properties for formal Laurent series and give some notations. Then, in Section 1.2, we will introduce Diophantine approximation and metric Diophantine approximation. This introduction will be split into homogeneous and inhomogeneous cases. We will collect some results for the two cases, especially the inhomogeneous case which consists of the so-called double-metric and single-metric cases. Finally, we will state the main goal of this thesis in Section 1.3. In Chapter 2, we will discuss the refinement of Kim, Nakada and Natsui. In Section 2.1, we will state some lemmas and present the proof of their refinement. In Section 2.2, we will give some improvements of the proofs from the previous section and use them to prove one direction of Kurzweil’s theorem. In Section 2.3, we will prove a special case of Kim, Nakada and Natsui’s refinement with a completely different method as the one in Section 2.1. In Chapter 3, we will show an analogue of the refinement proved by Kurzweil in [8]. Since this result and another refinement which was recently proved by Kim, Tan, Wang and Xu have some similarities and differences, we will compare them at the end of this chapter. Finally, in Chapter 4, we will end the thesis with some conjectures.