2005 Fiscal Year Final Research Report Summary
Markov spectrum in various Diophantine approximation problems
| Project/Area Number |
14540052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Suzuka National College of Technology |
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Principal Investigator |
YASUTOMI Shin-ichi Suzuka National College of Technology, General Education, Professor, 一般科目, 教授 (60230231)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Diophantine approximation / Markov spectrum / simultaneous approximation / Jacobi-Perron Algorithm |
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| Research Abstract |
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.
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