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Research for rational approximation to transcendental numbers

Research Project

Project/Area Number 16540149
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

HARA Masayoshi  Kyoto Univ., Graduate School of Science, Assistant Prof., 大学院・理学研究科, 助教授 (40156336)

Co-Investigator(Kenkyū-buntansha) YAMAGUCHI Masatoshi  Kyoto Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (30022651)
UEDA Tetsuo  Kyoto Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (10127053)
NAGATA Makoto  Kyoto Univ., RIMS, Assistant Professor, 数理解析研究所, 助手 (30293971)
Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsTranscendental numbers / Algebraic integers / Pisot numbers / Salem numbers / Uniform distribution / Rational approximation / Fractional parts / 有理近似 / Mahler測度 / 代表的整数 / Kronecker行列式
Research Abstract

The main purpose of this research was to study further the fractional parts of given increasing sequences done by, for example, Hardy, Gelfond, Thue, Siegel, Pisot and Salem, by using the method of rational approximation to transcendental numbers obtained by the head investigator for several years. One of the most interesting and important open problems may be the question whether there exists a transcendental number θ>1 for which the fractional parts of λθ^n,λ>0 converges to 0, as n tends to ∞, or not. It is already known that if such θ is algebraic, it must be a Pisot number, that is, a real algebraic integer greater than 1 whose other conjugates have modulus strictly less than 1. We could not solve this difficult problem unfortunately, however the head investigator succeeded to weaken the decay condition of the frational parts which implies that θ should be a Pisot number. This result was already published in Acta Arithmetica last year.
The smallness of the fractional parts means that the number is very close to some integer. From the point of view of rational approximation, this may produce a problem to find a good rational approximation of the form a_<n+1>/a_n to θ. And this observation will give an idea to use some analogue of Pade approximation to investigate this problem. However we could not realize this idea. This is still interesting, because we do not know any related results about transcendental numbers except for Boyd's result on badly approximated transcendental numbers. This may be a research on a sharp edge between algebraic and transcendental numbers. We will continue this research for a while. There is also a related interesting problem known as Mahler's 3/2 problem, which asks a lower bound of the fractional parts of the simple sequence (3/2)^n.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (8 results)

All 2005 2004

All Journal Article (8 results)

  • [Journal Article] Note on the fractional parts of λ θ ^n2005

    • Author(s)
      Masayoshi Hata
    • Journal Title

      Acta Arithmetica 120・2

      Pages: 153-157

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Note on the fractional parts of λθ^n2005

    • Author(s)
      M.Hata
    • Journal Title

      Acta Aritlimetica 120,no.2

      Pages: 153-157

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Note on the fractional part of λ θ^n2005

    • Author(s)
      Masayoshi Hata
    • Journal Title

      Acta Arithmetica 120・2

      Pages: 153-157

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Fixed points of polynomial automorphisms of C^n2004

    • Author(s)
      Tetsuo Ueda
    • Journal Title

      Avdanced Studies in Pure Math., Complex Analysis in Several Variables 42

      Pages: 319-324

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On diophantine approximations related to G-functions2004

    • Author(s)
      Makoto Nagata
    • Journal Title

      数理解析研究所 講究録 1274

      Pages: 94-99

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fixed points of polynomial automorphisms of C^n2004

    • Author(s)
      T.Ueda
    • Journal Title

      Advanced Studies in Pure Math., Complex Analysis in Several Variables 42

      Pages: 319-324

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On diophantine approximations related to G-functions2004

    • Author(s)
      M.Nagata
    • Journal Title

      Surikaisekikenkyusho Kokyuroku 1274

      Pages: 94-99

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fixed points of polynomial automorphisms of Cn,Complex Analysis in Serveral Variables2004

    • Author(s)
      T.Ueda
    • Journal Title

      Adv.Stud.Pure Math. 42

      Pages: 319-324

    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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