The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients (引用終わり)
On the complexity of algeraic numbers Iが(2007) On the complexity of algebraic numbers, IIが(2005)で、時間が逆転している
Annals of Mathの出版が遅いのか（＾＾ いや、ちょっと、連分数について調べているんだ（＾＾
http://adamczewski.perso.math.cnrs.fr/TMCF.pdf (avec Y. Bugeaud) Transcendence measure for continued fractions involving repetitive or symmetric patterns, J. Eur. Math. Soc. 12 (2010), 883--914.
http://adamczewski.perso.math.cnrs.fr/ Boris Adamczewski Institut Camille Jordan Universite Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex, France 0548現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/25(土) 21:10:43.16ID:QcNp0s4+>>545-546 おっちゃん、どうも、スレ主です。
(>>483より)＜再録＞ (抜粋) ２） 次に[HT09] より (抜粋) P3126 The derivation of this from our main result uses the upward topology on α in which, as we mentioned, the scattered sets are the finite subsets of α. A known result that we extend here is Theorem 5.1 from [5] in which the present is predicted from an “infinitesimal” piece of the past, and the predictor is correct except on a countable set that is nowhere dense. In terms of our framework here, we have the topology on R in which the basic open sets are half-open intervals (w, x] (so f ～x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [5].
[5] C. Hardin and A. Taylor, A peculiar connection between the axiom of choice and predicting the future, American Mathematical Monthly 115 (2008),
(>>484より)＜再録＞ (抜粋) ３） 最後に[（成書）The Mathematics of Coordinated Inference: A Study of Generalized Hat Problems]より (抜粋) P76 7.3 Corollaries
The second result we derive concerns the extent to which "the present can be predicted based on the past." Here, the exact characterization of the error sets occurs in Theorems 3.1 and 3.5 in [HT08b].
The derivation of this uses the topology on R in which the basic open sets are half-open intervals (w; x] (so f ～x g if f and g agree on (w, x) for some w < x). It is known that the scattered sets here are countable and nowhere dense. The exact characterization of the error sets in this example (as scattered sets) was absent in [HT08b].
With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict. After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be, “That is impossible. Given any strategy for predicting the values of an arbitrary function, ・・・” (引用終り)
だから、 この大げさな、”INTRODUCTION”にある [HT08b] でのこの大げさに書いた ”what the strategy would predict at t.”の部分が すっかり成書では、削除された。
そして、成書には、”7 The Topological Setting ・・・71”として７章でTopologicalな条件付きの議論になったわけ。 つまり、”With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict.”ではなく、繰返すが、Topologicalな条件付きの議論に後退したわけだ（＾＾
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)
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（＾＾
R - Dν≠｛x ｜(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x ｝ でもないんだ・・、多分（＾＾ 0579１３２人目の素数さん2017/11/26(日) 19:06:38.74ID:eS22cW4G ＞つまり、”Dν＝｛x ｜ for almost all x at Lemma 3 ｝”みたい（＾＾ アホ丸出しｗ 0580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) 19:08:40.25ID:1WQ1V5QH>>576 訂正
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！（＾＾ ↓ Dν≠｛x ｜(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.｝ なんだ！（＾＾
R - Dν≠｛x ｜(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x ｝ でもないんだ・・、多分（＾＾ ↓ R - Dν≠｛x ｜(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x ｝ なんだ・・、多分（＾＾
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（＾＾ http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf 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)
