現代数学の系譜工学物理雑談古典ガロア理論も読む49

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94: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/01(月) 23:35:35.73 ID:dCRrvhl7

>>93 つづき
Irrationality measure

The irrationality measure (or irrationality exponent or approximation exponent or Liouville?Roth constant) of a real number x is a measure of how "closely" it can be approximated by rationals. Generalizing the definition of Liouville numbers, instead of allowing any n in the power of q, we find the least upper bound of the set of real numbers μ such that

0<|x - p/q|< {1/q^μ

is satisfied by an infinite number of integer pairs (p, q) with q > 0. This least upper bound is defined to be the irrationality measure of x.[3]:246 For any value μ less than this upper bound, the infinite set of all rationals p/q satisfying the above inequality yield an approximation of x.
Conversely, if μ is greater than the upper bound, then there are at most finitely many (p, q) with q > 0 that satisfy the inequality; thus, the opposite inequality holds for all larger values of q. In other words, given the irrationality measure μ of a real number x, whenever a rational approximation x ? p/q, p,q ∈ N yields n + 1 exact decimal digits, we have

1/10^n >= |x - p/q| >= {1/q^(μ +ε)

for any ε>0, except for at most a finite number of "lucky" pairs (p, q).

For a rational number α the irrationality measure is μ(α) = 1.[3]:246 The Thue?Siegel?Roth theorem states that if α is an algebraic number, real but not rational, then μ(α) = 2.[3]:248

Almost all numbers have an irrationality measure equal to 2.[3]:246

Transcendental numbers have irrationality measure 2 or greater. For example, the transcendental number e has μ(e) = 2.[3]:185 The irrationality measure of π is at most 7.60630853: μ(log 2)<3.57455391 and μ(log 3)<5.125.[4]

The Liouville numbers are precisely those numbers having infinite irrationality measure.[3]:248
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1514376850/94

326: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/09(火) 23:29:48.09 ID:Xw3gWI4S

>>94
"Irrationality measure"について

https://en.wikipedia.org/wiki/Liouville_number
Liouville number
(抜粋)
6 Irrationality measure

0< |x-p/q|< 1/q^μ

is satisfied by an infinite number of integer pairs (p, q) with q > 0. This least upper bound is defined to be the irrationality measure of x.[3]:246
(引用終り)

http://mathworld.wolfram.com/IrrationalityMeasure.html
Irrationality Measure MathWorld Wolfram Research, Inc.

http://planetmath.org/irrationalitymeasure
irrationality measure planetmath.org Owner: mathcam Added: 2004-02-27 - 13:34 Author(s): mathcam Versions (v8) by mathcam 2013-03-22

(畑 政義先生)
https://projecteuclid.org/euclid.pja/1195511637
https://projecteuclid.org/download/pdf_1/euclid.pja/1195511637
Improvement in the irrationality measures of π and π^2 Masayoshi Hata Proc. Japan Acad. Ser. A Math. Sci. Volume 68, Number 9 (1992), 283-286.

https://www.math.kyoto-u.ac.jp/ja/people/profile/hata
畑 政義 京都大学 理学研究科／理学部 数学教室

http://rio2016.5ch.net/test/read.cgi/math/1514376850/326

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