The approach to number theory by erdadic theory

1999 Fiscal Year Final Research Report Summary

• Project/Area Number: 10640139
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Meijo University
• Principal Investigator: HARA-MIMACHI Yuko Meijo University, Faculty of Science and Technology, Lecturer, 理工学部, 講師 (00218629)
• Co-Investigator(Kenkyū-buntansha): SAITO Kimiaki Meijo University, Faculty of Science and Technology, Assistant Professor, 理工学部, 助教授 (90195983) ; KUBOTA Tomio Meijo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40022511)
• Project Period (FY): 1998 – 1999
• Keywords: ergadic theory / Diophantine approsimation / Simultaneous approximation

Research Abstract

In Diophantine approximation theory, we consider a conjective of Little-wood, that is, the simultaneous approximation problem for any n(n【greater than or equal】2) real numbers. It is known that this conjecture is true for n = 2. The purpose of this research is to discuss about this conjecture for two quadratic irrationals.
In 1998, we tried to approach by the method of H.Dickinson(1993,1994). This method is to combine a Diophantine inequality for the simultaneous or not simultaneous linear forms with the natural extension, skew product and substitution in ergodic theory. But it was failed.
In 1999, we tried to approach by using an analogue of the inequality of Littlewood conjecture, based on the Minkowski's convex body theorem. This inequality was shown by Minkowski and W, M..Schmidt, and improved by Cassels, Davenport and Mahler. By using This result and the periodicity of the expansions of two quadratic irrasionals, we may improve the inequality.