在正規 Laurent 級數下限制於兩個變數之賦矩 Diophantine 逼近

Abstract

在1988 年之前，許多研究者討論關於實數體下之賦距Diophantine 逼近在 Diophantine 不等式頂多限制一個變數。而從1988 年開始，英國數學家G. Harman 開始探討實數體下之賦距Diophantine 逼近在Diophantine 不等式限制兩個變數， 他希望可以得到是否有逼近的數學式能模擬解的個數，因此他對這個主題寫了四 篇的文章討論這個問題。而在這篇論文中，我們希望可以得到類似於G. Harman 的結果在正規Laurent 級數體下之賦距Diophantine 逼近。 本論文的主要架構如下：第一章，我們將回顧在實數下一些賦距Diophantine 逼近的一些結果。第一章節將分成兩小節，在第一小節中，我們將回顧在實數體 下一些基礎知識以及在賦距Diophantine 逼近在Diophantine 不等式限制一個變數 的一些前人所做的結果；在第二小節中，我們將介紹G. Harman 在實數體下之賦 距Diophantine 逼近針對在Diophantine 不等式限制兩個變數下的結果，以及推廣 他所得到的結論並應用在限制分子分母在某些條件下會得到是有限多個或是無窮 多個convergents 對於他的連分數展開，在這小節最後，我們將會提出這篇論文的 主要目的。 第二章，我們會介紹在正規Laurent 級數體下之賦距Diophantine 逼近的相關 背景知識以及一些過去在正規Laurent 級數體下之賦距Diophantine 逼近所得到的 結果。第二章節分成三小節，在第一小節中，我們將介紹一些背景知識關於正規 Laurent 級數體；在第二小節中，我們將介紹在正規Laurent 級數中連分數的相關 性質；在第三小節中，我們將介紹M. Fuchs 等人在正規Laurent 級數體下之賦距 Diophantine 逼近上的貢獻。 第三章，我們將提出我們對於正規Laurent 級數體下之賦距Diophantine 逼 近下所限制兩個變數的結果。第三章分成五小節，第一小節中我們討論在符合某 些條件下，我們將得到有限多個解在Diophantine 逼近不等式。在第二小節到第四 小節中，我們對於賦距Diophantine 逼近在Diophantine 不等式中限制兩個變數下 所得到的解的個數，其中第二小節是對分母是在positive lower asymptotic density 的集合裡的元素，分子是square-free 多項式；第三小節中是對於分母是在positive lower asymptotic density 的集合裡的元素，分子是在一個等差數列中的多項式；第 四小節是討論分母是不可分解的多項式，分子是在一個等差數列中的多項式。最 後在第五小節中將提出從第一小節到第四小節的一些應用在正規Laurent 級數體 下之連分數展開個數的結果。 在最後一章裡，我們將對這篇論文做一個總結，以及提出一些還未解決的問 題。

Before 1988, many researchers discussed metric Diophantine approximation in the field of real numbers with a restriction of only one variable in the Diophantine inequality. In 1988, a British mathematician, G. Harman, started to investigate metric Diophantine approximation in the field of real numbers with restrictions of two variables in the Diophantine inequality. G. Harman obtained zero-one laws and asymptotic formulas for the number of solutions of the Diophantine inequality with two restrictions. The purpose of the present work is to derive similar results for metric Diophantine approximation in the field of formal Laurent series. The main structure of this thesis is as follows: in Chapter 1, we will recall some results on metric Diophantine approximation in the field of real numbers. Chapter 1 is divided into two sections. In Section 1.1, we will review some basic knowledge on metric Diophantine approximation and review some results for Diophantine approximation with a restriction of one variable in the Diophantine inequality. In Section 1.2, we will introduce G. Harman’s results for Diophantine approximation with two restrictions in the Diophantine inequality in the field of real numbers. Moreover, we will also discuss some applications of Harman’s result which are concerned with whether or not there are finitely or infinitely many convergents with given restrictions for numerator and denominator. At the end of Section 1.2, we will explain in details the main goals of this thesis. In Chapter 2, we will summarize some known results on metric Diophantine approximation in the field of formal Laurent series. This chapter is divided into three sections. In Section 2.1, we introduce some preliminaries on the field of formal Laurent series. In Section 2.2, we discuss some properties of continued fractions in the field of formal Laurent series. In the last section, we will introduce some contributions on metric Diophantine approximation in the field of formal Laurent series, such as two results of M. Fuchs which will play a key role in this thesis. In Chapter 3, we will present some results with two restrictions for metric Diophantine approximation in the field of formal Laurent series. This chapter is divided into five sections. In Section 3.1, we give a necessary condition that there are finitely many solutions with two restrictions of the Diophantine inequality for metric Diophantine approximation in the field of formal Laurent series. In Section 3.2 to Section 3.4, we study the number of solutions with two restrictions in the infinite case. The first of these sections considers the restriction that the denominator is in a set with positive lower asymptotic density and the numerator is a square-free polynomial, the second section considers the restriction that the denominator is in a set with positive lower asymptotic density and the numerator is a polynomial in an arithmetic progression, and the third section considers the restriction that the denominator is ii an irreducible polynomial in an arithmetic progression and the numerator is in an arithmetic progression. In the last section of this chapter, we give some applications of the results from Section 3.1 to Section 3.4 to convergents. Finally, in the last chapter, we conclude the thesis by summarizing our results and by discussing possible directions of future research.