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 $(1/3, 1/sqrt{5}]$

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