[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 (658ﾚｽ)
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93(1): 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) 23:34:23.48 ID:dCRrvhl7(25/27) AAS
（参考）
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%82%A6%E3%83%B4%E3%82%A3%E3%83%AB%E6%95%B0
リウヴィル数
https://en.wikipedia.org/wiki/Liouville_number
(抜粋)
Structure of the set of Liouville numbers[edit]
For each positive integer n, set
U_n=∪_q=2〜∞ ∪_p= -∞ 〜∞ {x∈ R :0<|x - p/q|< 1/q^n｝ =∪_q=2〜∞ ∪_p= -∞〜∞ ( p/q - 1/q^n, p/q+ 1/q^n)＼ { p/q}
The set of all Liouville numbers can thus be written as
L=∩_n=1〜∞ U_n.
Each Un is an open set; as its closure contains all rationals (the p/q's from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, L is comeagre, that is to say, it is a dense Gδ set.
Along with the above remarks about measure, it shows that the set of Liouville numbers and its complement decompose the reals into two sets, one of which is meagre, and the other of Lebesgue measure zero.
つづく
94(1): 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(26/27) AAS
>>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)
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