正規Laurent級數體上探討Kurzweil定理

Abstract

我們將在本論文中探討正規Laurent級數體之下的賦距同步Diophantine逼近。在最近的一篇論文中，Kim和Nakada證明了在一維度的正規Laurent級數之下，和Kurzweil定理相似的一個結果。本論文主要的工作是提供一個新的證明方法，甚至可推廣到同步Diophantine逼近。 本文的主要架構如下：我們將在第一章介紹Diophantine逼近的背景。此章分為三小節。第一節，我們簡單地回顧Diophantine 逼近和賦距Diophantine逼近的概念，並說明一些在實數體下的結論及在正規Laurent 級數體下的相似結果。在第二節中，我們將介紹非齊次同步Diophantine逼近的概念。此外，我們羅列了一些定義和符號，以及關於所謂的double-metric和single-metric的結論。最後，第三節包含了我們本論文主要的結果。 在第二章，我們回顧一些在正規Laurent 級數體之下的基本性質。 而第三、四章包含了0-1 法則和一連串對於我們在一維、高維度的證明過程中非常重要的引理。而我們主要結果的證明就是根據這些引理得證。事實上，在一維度的結果即是高維度結論中的一個特例，但為了方便閱讀以及為了高維度證明的想法做準備，我們將優先處理一維度的例子。 最後，在第五章，我們將針對本論文做一個總結。

This thesis is concerned with metric simultaneous Diophantine approximation in the field of formal Laurent series. In a recent paper, Kim and Nakada proved an analogue of Kurzweil’s theorem in dimension one for formal Laurent series. The main aim of this thesis is to give a new proof which works for simultaneous Diophantine approximation as well. An outline of this thesis is as follows. In Chapter 1, we will introduce background on Diophantine approximation. This chapter is split into three sections. In Section 1.1, we will briefly recall Diophantine and metric Diophantine approximation, and state some results in the real case and some analogues over the field of formal Laurent series. Then, in Section 1.2, we will introduce inhomogeneous (simultaneous) Diophantine approximation. Moreover, we will collect notations and results for the so-called double-metric and single-metric cases. Finally, Section 1.3 will contain our main results. In Chapter 2, we will recall some fundamental properties for formal Laurent series. Chapter 3 and Chapter 4 will contain zero-one laws and a series of lemmas which are important for the proof of our results in dimension one and higher dimension, respectively. The proofs will follow from these lemmas. We want to point out that the result in dimension one is in fact only a special case of the higher dimensional result. Nevertheless, for the sake of readability and as a warm-up, we will treat the one-dimensional case separately. Finally, we will end the thesis with some concluding remarks in Chapter 5.