The construction of the infinite ergodic theory and the application to metric number theory

• Project/Area Number: 23740088
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Japan Women's University
• Principal Investigator: NATSUI Rie 日本女子大学, 理学部, 准教授 (60398633)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: エルゴード理論 / 測度論的数論 / ユークリッドアルゴリズム / 連分数変換 / 国際情報交換 / フランス / 韓国

Outline of Final Research Achievements

Toward a construction of a general system for the measurable dynamical system with the infinite invariant measure, this research tried to find the new interpretation about the randomness of number from the point of view of the ergodic theory. In particular, this research obtained some results for the complexity of number about number theoretic transformations, for example Euclidean algorithm in the positive characteristic theory, Farey maps on the Bruhat-Tits tree, and α-continued fraction transformations on a real number field, as the concrete research model.

Research Products

• [Journal Article] Farey Maps, Diophantine Approximation and Bruhat-Tits tree (2014)
  • Author(s): Dong Han Kim, Seonhee Lim, Hitoshi Nakada, and Rie Natsui
  • Journal Title: Finite Fields and Their Applications Volume: 30 Pages: 14-32
  • DOI: 10.1016/j.ffa.2014.05.007
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On the equivalence relations of α-continued fractions (2014)
  • Author(s): Hitoshi Nakada, Rie Natsui
  • Journal Title: Indagationes Mathematicae Volume: 25 Issue: 4 Pages: 800-815
  • DOI: 10.1016/j.indag.2014.02.006
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Fine costs for Euclid’s algorithm on polynomials and Farey maps (2014)
  • Author(s): Valerie Berthe, Hitoshi Nakada, Rie Natsui, Brigitte Vallee
  • Journal Title: Advances in Applied Mathematics Volume: 54 Pages: 27-65
  • DOI: 10.1016/j.aam.2013.11.001
  • Peer Reviewed
• [Journal Article] A refined Kurzweil type theorem in positive characteristic (2013)
  • Author(s): Dong Han Kim, Hitoshi Nakada, Rie Natsuic
  • Journal Title: Finite Fields and Their Applications Volume: 20 Pages: 64-75
  • DOI: 10.1016/j.ffa.2012.12.002
  • Peer Reviewed
• [Presentation] The equivalence relations of α-continued fraction
  • Author(s): Rie Natsui
  • Organizer: Measurable and Topological Dynamical Systems, Keio 2013
  • Place of Presentation: 慶應義塾大学
• [Presentation] On the existence of the Legendre constant for $\alpha$ continued fractions
  • Author(s): Rie Natsui
  • Organizer: Ergodic Theory and Metric Number Theory
  • Place of Presentation: 日本女子大学
• [Presentation] Bit complexity of some Euclidean type algorithms
  • Author(s): 夏井利恵
  • Organizer: エルゴード理論とその周辺
  • Place of Presentation: 大阪大学