Research for rational approximation to transcendental numbers

超越数有理逼近的研究

• 批准号: 16540149
• 负责人: HARA Masayoshi
• 金额: $ 1.41万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540149/
• 关键词: Transcendental 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

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.

本研究的主要目的是利用首席研究员多年来获得的超越数的有理逼近方法，进一步研究Hardy、Gelfond、Thue、Siegel、Pisot和Salem等人所做的给定递增序列的小数部分。最有趣和最重要的开放问题之一可能是是否存在一个超越数 θ>1，当 n 趋向于 无穷大时，λθ^n,λ>0 的小数部分收敛于 0。我们已经知道，如果这样的 θ 是代数的，那么它一定是一个皮索数，即大于 1 的实代数整数，而其他共轭的模严格小于 1。不幸的是，我们无法解决这个难题，但首席研究员成功地弱化了小数部分的衰减条件，这意味着 θ 应该是一个皮索数。这个结果去年已经发表在《算术学报》上。小数部分很小意味着这个数字非常接近某个整数。从有理逼近的角度来看，这可能会产生找到 a_<n+1>/a_n 到 θ 形式的良好有理逼近的问题。这一观察结果将给出一个使用 Pade 近似的类似物来研究这个问题的想法。然而我们无法实现这个想法。这仍然很有趣，因为除了博伊德关于严重近似超越数的结果之外，我们不知道任何关于超越数的相关结果。这可能是对代数数和超越数之间尖锐边缘的研究。我们将继续这项研究一段时间。还有一个相关的有趣问题，称为马勒 3/2 问题，它询问简单序列的小数部分的下界 (3/2)^n。

项目成果

Note on the fractional part of λ θ^n

注意 λ θ^n 的小数部分

• 发表时间: 2005
• 期刊: Acta Arithmetica 120・2
• 作者: S. Kanagawa;S. Ogawa;Masayoshi Hata
• 通讯作者: Masayoshi Hata

Note on the fractional parts of λ θ ^n

注意 λ θ ^n 的小数部分

• 发表时间: 2005
• 期刊: Acta Arithmetica 120・2
• 作者: 申正善;内藤敏機;Masayoshi Hata
• 通讯作者: Masayoshi Hata

Fixed points of polynomial automorphisms of C^n

C^n 多项式自同构的不动点

• 发表时间: 2004
• 期刊: Adv.Stud.Pure Math. 42
• 作者: Michihiko Fujii;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Masaaki Ue;Masaaki Ue;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Tetsuo Ueda
• 通讯作者: Tetsuo Ueda

On diophantine approximations related to G-functions

关于与 G 函数相关的丢番图近似

• 发表时间: 2004
• 期刊: 数理解析研究所 講究録 1274
• 作者: Toshiaki;Hishida;Yoshihiro;Shibata;Wataru Ichinose;Makoto Nagata
• 通讯作者: Makoto Nagata

Note on the fractional parts of λθ^n

注意 λθ^n 的小数部分

• 发表时间: 2005
• 期刊: Acta Aritlimetica 120,no.2
• 作者: 内藤敏機;申正善;M.Hata
• 通讯作者: M.Hata