現代数学の系譜工学物理雑談古典ガロア理論も読む46

[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46 (692ﾚｽ)
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575(2): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/26(日)18:47 ID:1WQ1V5QH(17/34) AAS
>>398 補足

戻る
外部ﾘﾝｸ[pdf]:www.unirioja.es
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009

fν(x)
＝0　if x ∈ R - Q（無理数）
＝1/q^ν if x = p/q ∈ Q, irreducible （有理数で既約分数）
で

Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not differentiable) at the rationals, and continuous at the irrationals.
With respect the differentiability, we have:
(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.
Moreover, the sets of numbers that fulfill (a) and (b) are both of them un-countable.
(引用終り)

ここ、無理数を
(a) For every irrational number x with bounded elementsと、
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.と
完全に２分したと読んだので、あとの測度論の下記Theorem 2

P6
Theorem 2. For ν > 2, let us denote
Cν = {f ∈ R : fν is continuous at x }
Dν = {f ∈ R : fν is differentiable at x }

Then, the Lebesgue measure of the sets R - Cν and R - Dν is 0, but the four
sets Cν, R - Cν, Dν, and R - Dν are dense in R.
(引用終り)

つづく

576(3): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/26(日)18:48 ID:1WQ1V5QH(18/34) AAS
>>575 つづき

これと、
外部ﾘﾝｸ[pdf]:argent.shinshu-u.ac.jp
A. Ya. ヒンチン（Khinchin）著　連分数（訳：乙部厳己）1961
P58
定理29. (0, 1) の中の有界な要素をもつ数の全体は測度0 である。
(引用終り)

とが整合しないので、いろいろ調べていたんだ（>>556とか）（＾＾
ようやく分ったのは、
Dν≠｛x ｜(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.｝
じゃないんだ！（＾＾

VARONA氏のP5 Lemma 3 g(t)について示しているように、”for almost all x”がDνなんだ。
つまり、”Dν＝｛x ｜ for almost all x at Lemma 3 ｝”みたい（＾＾

上記の”(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.”は、こんなのもあると、一例を示したと
１週間近く悩んでいたんだ（＾＾

以上

596(1): 現代数学の系譜雑談古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/26(日)23:26 ID:1WQ1V5QH(31/34) AAS
>>575 補足

原本PDFを見て貰った方が視認性は良いが、後の検索性のためにコピペする（＾＾
外部ﾘﾝｸ[pdf]:www.unirioja.es
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009
(抜粋)
P7
4. The theorem of Thue-Siegel-Roth revisited

Or, equivalently, if x is an irrational algebraic number, there exists a positive constant C(x, α) such
that ｜x - p/q ｜< C(x, α)/q^(2+α) (10)
has no rational solution.

P8
Remark 3. We have proved Theorem 3 by using the Thue-Siegel-Roth theorem.
But we have said that it is a reformulation. So, let us see how to
deduce the Thue-Siegel-Roth theorem from Theorem 3.
Given x algebraic and irrational, and ν > 2, Theorem 3 ensures that fν
is differentiable at x, so there exists
lim y→x {fν(y) - fν(x)}/(y - x) = f’ν (x).
By approximating y → x by irrationals y, it follows that f’ν (x) = 0.
Consequently, by approximating y → x by rationals, i.e., y = p/q, we also must have
lim p/q→x ｛fν(p/q) - fν(x)｝/(p/q - x ) = lim p/q→x (1/qν)/(p/q - x) = 0.
Then, for every ε > 0, there exists δ > 0 such that
1/(q^ν) <= ε｜p/q - x｜
when p/q ∈ (x - δ, x + δ). From here, it is easy to check that the same
happens for every p/q ∈ Q, perhaps with a greather constant ε' in the place
of ε. Thus, (10) with α = ν-2 and some positive constant C(x, α) = 1/ε' has
no rational solution, and we have obtained the Thue-Siegel-Roth theorem.
(引用終り)

つづく

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