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On the multiple recurrence of infinite measure preserving transformations and a conjecture by Erdos

Research Project

Project/Area Number 16K13766
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionKeio University

Principal Investigator

Nakada Hitoshi  慶應義塾大学, 理工学部(矢上), 名誉教授 (40118980)

Research Collaborator Aaronson Jon  Tel Aviv University, Faculty of Exact Sciences, Professor
Sarig Omri  Weizmann Institute of Science, Faculty of Mathematics and Computer Science, Professor
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywordsエルゴード理論 / 無限大不変測度 / 等差数列のエルデシ予想 / 等差数列 / エルデシ予想 / 無限エルゴード理論 / 多重再帰性 / 連分数変換 / 不変測度 / 解析学 / 数論
Outline of Final Research Achievements

We consider the following the long standing open question which is called the Erdos conjecture for arithmetic progressions : Suppose that (a_n) is a sub-sequence of natural numbers such that the sum of the inverse 1/a_n diverges. Then for any natural number k, there exists an arithmetic progression of length k in (a_n). The aim of this research is to find a way to solve this conjecture from infinite ergodic theory. In this point of vie w, we got the following results. (1) We constructed the natural extension of the Rauzy induction as a map on the set of translation surfaces. (2) We have some limit theorems for cylinder flows.
Moreover we constructed an infinite measure preserving transformations which has a restricted mutiplicity of recurrence.

Academic Significance and Societal Importance of the Research Achievements

等差数列に関するエルデシ予想は21世紀に入り、Green-Taoにより素数列に関しては解決されたものの、本来の問題は依然として未解決の難問である。本課題では、この問題解決への一つのアプローチとして1970年代に H. Furstenberg により提案された方法の厳密な正しさを証明することを意識しながら infinite ergodic theory を研究した。infinite ergodic theory の多重再帰性に関する研究の進展はエルデシ予想の解決に向けた一つの大きな可能性を持つもので、そこに本研究の学術的意義が見いだされる。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (24 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (9 results) Journal Article (3 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (11 results) (of which Int'l Joint Research: 8 results,  Invited: 9 results) Funded Workshop (1 results)

  • [Int'l Joint Research] テルアビブ大学/ヘブライ大学/ワイツマン研究所(イスラエル)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] パリ第7大学(フランス)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] デルフト工科大学(オランダ)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] テルアビブ大学/ヘブライ大学/ワイツマン研究所(イスラエル)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] デルフト工科大学(オランダ)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] パリ第7大学(フランス)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] Tel Aviv University(Israel)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Williams College(米国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Delft University of Technology(オランダ)

    • Related Report
      2016 Research-status Report
  • [Journal Article] On the construction of the natural extension of the Hurwitz complex continued fraction map.2019

    • Author(s)
      Hiromi Ei, Shunji Ito, Hitoshi Nakada, Rie Natsui
    • Journal Title

      Monatshefte fur Mathematik

      Volume: 188 Issue: 1 Pages: 37-86

    • DOI

      10.1007/s00605-018-1229-0

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A Piecewise Rotation of the Circle, IPR Maps and Their Connection withTranslation Surfaces2018

    • Author(s)
      Kae Inoue、Hitoshi Nakada
    • Journal Title

      Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics, Lecure Note in Mathematics

      Volume: 2213 Pages: 393-431

    • DOI

      10.1007/978-3-319-74908-2_19

    • ISBN
      9783319749075, 9783319749082
    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Discrepancy Skew Products and Affine Random Walks2017

    • Author(s)
      Jon Aaronson, Michael Bromberg, Hitoshi Nakada
    • Journal Title

      Israel Journal of Mathematics

      Volume: -

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] On the notions of suborbital graph and geodesic continued fractions for an imaginary quadratic field2019

    • Author(s)
      Hitoshi Nakada
    • Organizer
      FWF/JSPS Meeting at St. Virgil
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] On the Construction of Translation Surfaces from Piecewise Rotation Maps of the Circle2018

    • Author(s)
      Hitoshi Nakada
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Ergodic theory of complex continued fraction maps2018

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Workshop「60 years of dynamics and number expansions」
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The visits to zero of a cylinder flow2018

    • Author(s)
      Hitoshi Nakada
    • Organizer
      St Virgil FWF/JSPS Meeting Salzburg
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On Cruz and da Rocha's induction for piecewise rotation maps of the circle as the dual of Rauzy induction2017

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Modern Problems of Dynamical Systems and Their Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A construction of translation surfaces based on Cruz and da Rocha's idea2017

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Aperiodic patterns in Crystals, Numbers and Symbols
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On Cruz and da Rocha's idea for a piecewise rotation map of the circle and its application2017

    • Author(s)
      Hitoshi Nakada
    • Organizer
      Zero Entropy System
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Extended Rauzy induction の引き起こすRauzy classes 間のtransition について2017

    • Author(s)
      仲田 均
    • Organizer
      数論とエルゴード理論
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県金沢市)
    • Related Report
      2016 Research-status Report
  • [Presentation] Basic ideas in innite ergodic theory(I), (II), (III)2016

    • Author(s)
      仲田 均
    • Organizer
      エルゴード理論の最近の発展
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2016-10-19
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Szemerédiの定理と展望1、22016

    • Author(s)
      仲田 均
    • Organizer
      2016年度八王子数論セミナー
    • Place of Presentation
      八王子セミナーハウス(東京都八王子市)
    • Year and Date
      2016-08-27
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] On the natural extensions of some complex continued fraction transformations2016

    • Author(s)
      仲田 均
    • Organizer
      Ergodic Theory and its Connections with Arithmetic and Combinatorics
    • Place of Presentation
      Marseille(France)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Recent Progress in Ergodic Theory2018

    • Related Report
      2018 Annual Research Report

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Published: 2016-04-21   Modified: 2022-02-22  

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