| Budget Amount *help |
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2003: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
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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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