2005 Fiscal Year Final Research Report Summary
Research for rational approximation to transcendental numbers
| Project/Area Number |
16540149
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
HARA Masayoshi Kyoto Univ., Graduate School of Science, Assistant Prof., 大学院・理学研究科, 助教授 (40156336)
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| 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)
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| Project Period (FY) |
2004 – 2005
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| Keywords | Transcendental numbers / Algebraic integers / Pisot numbers / Salem numbers / Uniform distribution / Rational approximation / Fractional parts |
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| 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.
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