Gauss映射的自然扩张与Hermite最佳逼近
The natural extension of the Gauss map and Hermite best approximations
摘要：
Lagarias引入了实数$θ$的Hermite最佳逼近向量。一个非零向量（p，q）$\in$ Z × N是$θ$的Hermite最佳逼近向量，如果存在$Δ$ > 0使得对于所有非零（a，b）$\in$ Z 2，（p-qPDMATHABMM）2 + q2/$Δ$ $\le$（a-bPDMATHABMM）2 + b2/ $Δ$. Hermite观察到，如果q > 0，那么分数p/q必须是$θ$的连分数展开的收敛，Lagarias指出，一些收敛与Hermite最佳逼近向量无关。本文证明了Hermite最佳逼近向量在收敛向量中的几乎必然比例为ln 3/ ln 4。证明的主要工具是高斯映射x $\in$ ]0，1[ $\rightarrow$ {1/x}的自然扩张.
英文摘要：
Hermite best approximation vectors of a real number $θ$ were introduced by Lagarias. A nonzero vector (p, q) $\in$ Z x N is a Hermite best approximation vector of $θ$ if there exists $Δ$ > 0 such that (p -- q$θ$) 2 + q 2 /$Δ$ $\le$ (a -- b$θ$) 2 + b 2 /$Δ$ for all nonzero (a, b) $\in$ Z 2. Hermite observed that if q > 0 then the fraction p/q must be a convergent of the continued fraction expansion of $θ$ and Lagarias pointed out that some convergents are not associated with a Hermite best approximation vectors. In this note we show that the almost sure proportion of Hermite best approximation vectors among convergents is ln 3/ ln 4. The main tool of the proof is the natural extension of the Gauss map x $\in$]0, 1[$\rightarrow$ {1/x}.
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