Recent progress in ergodic theory of number theoretic algorithms

2015 Fiscal Year Final Research Report

• Project/Area Number: 24340020
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Keio University
• Principal Investigator: Nakada Hitoshi 慶應義塾大学, 理工学部, 教授 (40118980)
• Co-Investigator(Kenkyū-buntansha): ATSUJI Atsushi 慶應義塾大学, 経済学部, 教授 (00221044) ; TAMURA Yozo 慶應義塾大学, 理工学部, 教授 (50171905) ; TAMURA Akihisa 慶應義塾大学, 理工学部, 教授 (50217189) ; FUKUYAMA Kstusi 神戸大学, 大学院理学研究科, 教授 (60218956) ; NATSUI Rie 日本女子大学, 理学部, 准教授 (60398633) ; HAMA Masaki 文京学院大学, 外国部学部, 准教授 (90389875) ; INOUE Kae 慶應義塾大学, 薬学部, 講師 (90621011) ; ISHIKAWA Shiro 慶應義塾大学, 理工学部, 非常勤講師 (10051913)
• Co-Investigator(Renkei-kenkyūsha): MORITA Takehiko 大阪大学, 大学院理学研究科, 教授 (00192782) ; EI Hiromi 弘前大学, 大学院理工学研究科, 助教 (60333051)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: エルゴード理論 / 連分数 / ユークリッドアルゴリズム / 一様分布論

Outline of Final Research Achievements

The research on ergodic theory has had a strong relation to the theory of numbers even from the beginning until now. In particular, there are a number of open questions in the theory of binary, decimal, and continued fraction expansions. On the other hand, there are also new open questions concerning the theory of cryptography and the communication theory. In the present research, we mainly study the approximation theory of continued fractions with its speed of the convergence and also study the probabilistic estimate of the calculation costs of the Euclidean algorithms. We have some new results concerning these problems as well as other related subjects as the theory of uniform distributions.

Free Research Field

数理科学