Diophantine approximation and the Markoff chain

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/7878

Title:	Diophantine approximation and the Markoff chain
Authors:	黎右強;You-Chiang Li
Contributors:	數學研究所
Keywords:	馬可夫鏈;Markoff chain;Diophantine approximation
Date:	2006-06-28
Issue Date:	2009-09-22 11:07:55 (UTC+8)
Publisher:	國立中央大學圖書館
Abstract:	對於實數$xi$我們定義$\|\|xi\|\|$為最接近$xi$整數。我的論文主要是探討$V={liminf_{qin mathbb{N}}q\|q xi \|:xi in mathbb{R} setminusmathbb{Q}}.$ 這個集合。此篇論文裡面有三個重要定理，分別是Dirichlet、Hurwitz和Markoff的定理。由Dirichlet的定理我們可證得 $Vsubset[0,1]$。而由Hurwitz的定理，我們更進一步推得 $Vsubset[0,1/sqrt{5}]$，並且$1/sqrt{5}$ 將不能再更小。Markoff的定理則是一個重要的結果，他清楚的說明了集合$V$在 $(1/3, 1/sqrt{5}]$ 這個區間上分布的情形。 For raal $xi$, we define $\|\|xi\|\|$ be the nearest integer. We are interested in the set $V={liminf_{qin mathbb{N}}q\|q xi \|:xi in mathbb{R} setminusmathbb{Q}}.$ . Our main theorems are the Dirichlet's theorem, the Hurwitz's theorem and the Markoff's theorem. From Dirichlet’s theorem, we may prove that $Vsubset[0,1]$. And from Hurwitz’s theorem, we may obtain that $Vsubset[0,1/sqrt{5}]$ and $1/sqrt{5}$ cannot be improved. Markoff's theorem is an important result. He explained how $V$ distributes over the interval $(1/3, 1/sqrt{5}]$
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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