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Markov spectrum in various Diophantine approximation problems

各种丢番图近似问题中的马尔可夫谱

基本信息

批准号：
14540052
负责人：
YASUTOMI Shin-ichi
金额：
$ 0.96万
依托单位：
Suzuka National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

Let α,β be real cubic numbers andβ∈Q(α) and (α,β) be periodic point related to modified Jacobi-Perron Algorithm with the period k. We suppose that Q(α) is not totally real field. Jointing work with Ito Shunji(Kanazwa Univ) and Furukado Maki(Yokohama National Univ) we establish that the intermediate convergents associated with modified Jacobi-Perron Algorithm give the best rational approximation to (α,β).Our main Theorem is as follows :Main Theorem. Let (α,β) be periodic point related to modified Jacobi-Perron Algorithm with the period k and Q(α) be not totally real cubic field. The convergents associated with modified Jacobi-Perron Algorithm did not always give the best rational approximation. But, there exists sequnces of integers a_1,a_2,...,a_j and m_1,m_2,...,m_j such that {a_1q_<kn+m_1>+a_2q_<kn+m_2>+...+a_jq_<kn+m_1>} give the best rational approximation, that is, we denote Q_n=a_1q_<kn+m_1>+a_2q_<kn+m_2>+...+a_jq_<kn+m_j>. P_n and R_n are denoted by the nearest integer to Q_nα and Q_nβ respectively. Then, the limit set of √<Q_n>(Q_nα-P_n,Q_nβ-R_n) as n→∞ is the nearest ellipse to the origin.We introduce new algorithm for inhomogeneous Diophantine approximation and we obtain some good properties of this algorithm.

设α，β为真实的立方数，β∈Q（α），（α，β）为周期为k的修正Jacobi-Perron算法的周期点.我们假设Q（α）不是全真实的域。与Ito Shunji（Kanazwa Univ）和Furukado Maki（Yokohama National Univ）合作，我们建立了修正Jacobi-Perron算法的中间收敛性给出了（α，β）的最佳有理逼近。设（α，β）为周期为k的修正Jacobi-Perron算法的周期点，Q（α）为不完全真实的三次域.修正的Jacobi-Perron算法的收敛性并不总是给出最佳的有理逼近。但是，存在整数序列a_1，a_2，. a_j和m_1，m_2，.，m_j使得{a_1q_<kn+m_1>+a_2q_<kn+m_2>+. a_jq_<kn+m_1>}给出了最佳有理逼近，即记Q_n=a_1q_<kn+m_1>+a_2q_<kn+m_2>+. a_jq_<kn+m_j>。P_n和R_n分别用与Q_nα和Q_nβ最接近的整数表示。然后，证明了当<Q_n>n→∞时，n（Q_nα-P_n，Q_nβ-R_n）的极限集是离原点最近的椭圆，给出了非齐次丢番图逼近的新算法，并得到了该算法的一些好的性质。