High level people は自分達で勝手に立てたスレ28へどうぞ！sage進行推奨（＾＾； また、スレ43は、私が立てたスレではないので、私は行きません。そこでは、私はスレ主では無くなりますからね。このスレに不満な人は、そちらへ。 http://rio2016.2ch.net/test/read.cgi/math/1506152332/ 旧スレが512KBオーバー（又は間近）で、新スレ立てる （スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。）
いやはや、（文系） High level people たち（ ID:jEMrGWmk さん含め）の、数学ディベートもどきは面白いですね（＾＾； ”手強い？”とは・・、まさに、ディベートですね
私ら、理系の出典（URL)とコピペベース、ロジック（論証）＆証明重視のスタンスと、ディベートもどきスタイル（２ＣＨスタイル？）とは、明白に違いますね 私ら、（文系） High level people たちとの議論は、時間とスペースの無駄。レベルが高すぎてついていけませんね。典拠もなしによく議論しますね。よく分かりましたよ（＾＾；
もし張り付けによって反例を作りたいのなら、x＝0 の近傍が「離散的に分布する」ような貼り付けでは意味が無いんだぞ？ なぜなら、それ以外の開区間を取れば、そこではリプシッツ連続になるからだ。従って、張り付けによって反例を作りたいのなら、 x＝0 の近傍が「 R の中に稠密に分布する」ような貼り付けを考えなければならないんだぞ？
では、一例として、関数 f(x) を有理数 p だけ平行移動した f(x＋p) という関数を考え、これらを単純に足し算した
g(x) ＝ Σ[p∈Q] f(x＋p)
という関数を考えてみよう。この場合、"x＝0 の近傍" の挙動をする点が R の中に稠密に分布するように見えるが、 まず大前提として、上記のように定義した g は「ちゃんと各点で収束しているのか？」という問題が生じる。 もし収束してないなら、この g はそもそも well-defined でないことになるので失敗である。また、 仮に収束しているのだとしても、今度は
―――――――――――――――――――――――――――――――――――――――――――――――― 証明その２： f は点 x で微分可能とする。ケース１,２に場合分けすることで、f が点 x で連続であることを導く。
ケース１： f は点 x で連続であると仮定する。よって、f は点 x で連続である。
ケース２： f は点 x で連続でないと仮定する。一方で、lim[y→x](f(y)－f(x))/(y―x) ＝ f'(x) が存在するのだったから、 lim[y→x](f(y)－f(x)) ＝ lim[y→x](f(y)－f(x))/(y－x) * (y－x) ＝ f'(x) * 0 ＝ 0 となる。 すなわち、lim[y→x] f(y)＝f(x) となる。よって、f は点 x で連続である。これは、 f が点 x で連続でないという仮定に矛盾する。矛盾した状態からはどんな条件も導けるので、 特に、「 f は点 x で連続である」という条件が導ける。
よって、いずれのケースにおいても、f は点 x で連続であることが言えた。 ――――――――――――――――――――――――――――――――――――――――――――――――
上記の証明は、 ―――――――――――――――――――――――――――― 「 P → Q 」という形の命題を証明するのに必要なのは、 「 P を仮定すれば Q が導ける」ことを示すことだけである ――――――――――――――――――――――――――――
―――――――――――――――――――――――――――――――――――――――――――――――― 証明その３： f は点 x で微分可能とする。ケース１,２に場合分けすることで、f が点 x で連続であることを導く。
ケース１： f は点 x で連続であると仮定する。よって、f は点 x で連続である。
ケース２： f は点 x で連続でないと仮定する。一方で、lim[y→x](f(y)－f(x))/(y―x) ＝ f'(x) が存在するのだったから、 lim[y→x](f(y)－f(x)) ＝ lim[y→x](f(y)－f(x))/(y－x) * (y－x) ＝ f'(x) * 0 ＝ 0 となる。 すなわち、lim[y→x] f(y)＝f(x) となる。よって、f は点 x で連続である。
よって、いずれのケースにおいても、f は点 x で連続であることが言えた。 ――――――――――――――――――――――――――――――――――――――――――――――――
上記の証明は、本質的には「その２」と全く同じであり、 ―――――――――――――――――――――――――――― 「 P → Q 」という形の命題を証明するのに必要なのは、 「 P を仮定すれば Q が導ける」ことを示すことだけである ――――――――――――――――――――――――――――
という原則に立ち返った証明である。ケース２では やはり矛盾が起きているが、もはや
「矛盾した状態からは何でも帰結できるので、～～～」
といった言い回しすらない。それもそもはず、その言い回しをするより前に、示したかった条件「 f は点 x で連続である 」が 導けているからだ。上記の原則に立ち返った場合、「 Q 」が導けた時点で、それ以上何も言う必要がないので、 「矛盾した状態からは何でも帰結できるので、～～～」という言い回しすらしていないのが、この証明である。
まだ、疑問に思っているのは 下記のDifferentiability of the Ruler Functionの記述と貴方の定理との整合性だ
http://mathforum.org/kb/message.jspa?messageID=5432910 （>>35より） Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007 (抜粋) The ruler function f is defined by f(x) = 0 if x is irrational, f(0) = 1, and f(x) = 1/q if x = p/q where p and q are relatively prime integers with q > 0.
Using ruler-like functions that "damp-out" quicker than any power of f gives behavior that one would expect from the above.
Let w:Z+ --> Z+ be an increasing function that eventually majorizes every power function. Define f_w(x) = 0 for x irrational, f_w(0) = 1, and f_w(p/q) = 1/w(q) where p and q are relatively prime integers.
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)
Interesting, each of the sets of points where these functions fail to be differentiable is large in the sense of Baire category.
THEOREM: Let g be continuous and discontinuous on sets of points that are each dense in the reals. Then g fails to have a derivative on a co-meager (residual) set of points. In fact, g fails to satisfy a pointwise Lipschitz condition, a pointwise Holder condition, or even any specified pointwise modulus of continuity condition on a co-meager set.
(Each co-meager set has c points in every interval.)
[13] Gerald Arthur Heuer, "Functions continuous at irrationals and discontinuous at rationals", abstract of talk given 2 November 1963 at the annual fall meeting of the Minnesota Section of the MAA, American Mathematical Monthly 71 #3 (March 1964), 349.
The complete text of the abstract follows, with minor editing changes to accommodate ASCII format.
Earlier results of Porter, Fort, and others suggest additional questions about the functions in the title. Differentiability and Lipschitz conditions are considered. Special attention ispaid to the ruler function (f) and its powers. Sample results: THEOREM: If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere differentiable, but is Lipschitzian on a dense subset of the reals. THEOREM: If r > 0, f^r is continuous but not Lipschitzian at every Liouville number; if r > 2, f^r is differentiable at every algebraic irrational. THEOREM: If g is continuous at the irrationals and not continuous at the rationals, then there exists a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO: The last theorem follows from the following stronger and more general result. Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below. (引用終り)
ああ、いま改めて読むと Bulletin of the Calcutta Mathematical Society 49 (1957) Senguptaより ”・・・ f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set).”
なんてありますね。”at least one of the four Dini derivates of f is infinite”が、貴方の定理に近いかな？ ”Then E is co-meager in R (i.e. the complement of a first category set).”か・・ これか、これに近い文献を読まないことには、訳わからんな
えーと、Meagre setか・・ ”E is co-meager in R”が、イメージできんな・・（＾＾
前提a)(連続不連続が稠密)を、b)(連続とディニ微分発散が稠密な組み合わせ)に、緩和しても・・ a) f is continuous and discontinuous are each dense in R. ↓ b) f is continuous and the E *) are each dense in R. ( *)the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.)
Examples Subsets of the reals The rational numbers are meagre as a subset of the reals and as a space ? that is, they do not form a Baire space. The Cantor set is meagre as a subset of the reals, but not as a space, since it is a complete metric space and is thus a Baire space, by the Baire category theorem.
まず、関連参考：検索でヒットしたので貼る。 BaireCategory.pdfの”3. Pointwise limits of continuous functions.”に、「422に書いた定理」の関連記述 「Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous on a dense subset of R.)」とあり（当たり前か？ （＾＾ ）
http://www.math.utk.edu/~freire/teaching/m447f16/m447f16index.html MATH 447- Advanced Calculus I- Fall 2016- A. FREIRE (or: ANALYSIS IN R^n) (抜粋) http://www.math.utk.edu/~freire/teaching/m447f16/BaireCategory.pdf Sets of discontinuity and Baire's theorem Baire Category Notes (5 problems) (the problems are HW8, due Friday 11/4)A. FREIRE 2016 (抜粋) 1. Sets of discontinuity. For f : R → R, we define Df = {x ∈ R; f is not continuous at xg:
3. Pointwise limits of continuous functions. Theorem. If f : R → R is a pointwise limit of continuous functions, then Df is Fσ meager (that is, a countable union of closed sets with empty interior). (In particular, by Baire's theorem, f is continuous on a dense subset of R.)
Proof. We know Df = ∪ n>=1 D1/n (see Section 1), so it suffices to show that the closed sets Dε have empty interior, for any ε > 0. By contradiction, suppose Dε contains an open interval I. We'll find an open interval J ⊂ I disjoint from Dε! Let fn → f pointwise on R, with each fn : R → R continuous. For each N >= 1, consider the set: CN = {x ∈ I; (∀m, n >= N)|fm(x) - fn(x)| <= ε/3}. Clearly ∪ N>=1 CN = I (by pointwise convergence). QED (引用終り)
http://www.math.utk.edu/~freire/teaching/m447f16/AscoliArzelaNotes.pdf Ascoli-Arzela-Notes (final-included 7 exercises with solutions, and 11 extra problems.)
http://www.math.utk.edu/~freire/teaching/ Alex Freire Department of Mathematics University of Tennessee （終り）
あと、いま、「422に書いた定理」に、似た文献を見つけて読んでいる。（＾＾ ”I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994” これ、出版されていて、アマゾンでもヒットした
疑問が二つ １）Proposition 1.1.1. の「Given ε > 0 there is a δ > 0 such that {x ∈ (x0 － δ, x0 + δ) : |f(x) － f(x0)| ≧ ε} ∈ J.」で、普通のεδ論法だと、 |f(x) － f(x0)| ＜ ε と不等号の向きが逆になると思うが、誤植か？ σ-ideal を考えているから、これで良いのか？ どうも良いみたいだが
２）Corollary 1.1.6. の「(ii): There exists a residual set K such that f|K is continuous.*2」で、f|Kは、Theorem 1.1.4.の”(ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.”の記載ぶりとの比較から、f|Kcの誤記かなと思ったり？ 意味が全く違ってくる
http://www.math.wvu.edu/~kcies/prepF/BookIdensity/BookIdensity.pdf I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994- 被引用数: 84 (抜粋) CHAPTER 1 The Ordinary Density Topology 1.1. A Simple Category Topology
To gain some insight into what is happening with limits like this, it is useful to generalize this idea to a topological setting. A nonempty family J ⊂P(X) of subsets of X is an ideal on X if A ⊂ B and B ∈ J imply that A ∈ J and if A∪B ∈ J provided A,B ∈ J. An ideal J on X is said to be a σ-ideal on X if ∪n∈N An ∈ J for every family {An : n ∈ N} ⊂ J. Let J be an ideal on R and Ｔo be the ordinary topology on R. The set T (J) = {G ＼ J : G ∈ Ｔo, J ∈ J} is a topology on R which is finer than Ｔo. The following proposition is evident from the definitions.
Proposition 1.1.1. Let J be a σ-ideal on R and T (J ) be as above. For f : (R, T (J )) → (R, Ｔo) and x0 ∈ R the following statements are equivalent to each other. (i): f is continuous at x0. (ii): Given ε > 0 there is a δ > 0 such that {x ∈ (x0 － δ, x0 + δ) : |f(x) － f(x0)| ≧ ε} ∈ J.
Theorem 1.1.4. Let J be a σ-ideal and f : R → R. The following statements are equivalent. (i): The function f is J -continuous J -a.e. (ii): There exists a set K ∈ J such that the restricted function f|Kc is continuous.
Furthermore, if the ideal J contains no interval, then the following statement is equivalent to (i) and (ii) (iii): There exists a function g : R → R such that f = g J -a.e. and g is continuous in the ordinary sense J -a.e.
Proof. The fact that (ii) implies (i) is obvious. Suppose (i) is true and let f be J -continuous on a set M = Jc for J ∈ J. For each n ∈ N and x ∈ M, by Proposition 1.1.1(ii) there is an open interval I(n, x) and a J(n, x) ∈ J such that x ∈ I(n, x) \ J(n, x) ⊂ f－1((f(x) － 1/n, f(x) + 1/n)). For each fixed n, there must be a countable sequence xn,m ∈ M such that M ⊂∪m∈N I(n, xn,m). Let K = J ∪ ∪ n,m∈N J(n, xn,m) ∈ J. If x ∈ Kc and ε > 0, then there must exist natural numbers n and m such that 2/n < ε and x ∈ I(n, xn,m). Then |f(x) － f(xn,m)| < 1/n so that f(x) ∈ (f(xn,m) － 1/n, f(xn,m) + 1/n) ⊂ (f(x) － ε, f(x) + ε)
and I(n, xn,m) ∩ Kc ⊂ f－1((f(x) － ε, f(x) + ε)). Hence, f|Kc is continuous at x.
To prove the last part of the theorem, note first that (iii) implies (ii) even without the restriction that J contains no interval. Now suppose that J contains no interval and that f,K are as in (ii). Define (1) G(x) = lim sup t→x,t∈Kc f(t) and (2) g(x) = G(x) when G(x) is finite, or = f(x) otherwise. In particular, it follows from (ii) that f|Kc = g|Kc . Let x ∈ Kc and ε > 0. According to (ii) there is a δ > 0 such that (3) |g(y) － g(x)| = |f(y) － f(x)| < ε/2 whenever y ∈ (x － δ, x + δ) ∩Kc. If z ∈ (x － δ, x + δ) ∩K, then the assumption that K can contain no nonempty open set implies the existence of a sequence {zn : n ∈ N} ⊂ (x － δ, x + δ) ∩ Kc such that f(zn) → G(z). Hence, by (3), G(z) is finite, so g(z) = G(z) and |g(z) － g(x)| ? ε/2 < ε. Therefore, g is continuous at x. QED
The following example is interesting in light of the previous theorem.
Example 1.1.5. Let I be the σ-ideal consisting of all first category subsets of R. I-continuity is often called qualitative continuity [26]. It is well-known in this case that f is a Baire function if, and only if, f is qualitatively continuous I-a.e.
In particular, combining Example 1.1.5 with Theorem 1.1.4 yields the following well-known corollary, which will be useful in the sequel.
Corollary 1.1.6. Let f : R → R. The following statements are equivalent. (i): f is a Baire function. (ii): There exists a residual set K such that f|K is continuous.*2 (iii): f is qualitatively continuous I-a.e. In the case of Lebesgue measure, the following is true.
*2 A set is residual if its complement is first category. This is often called comeager.
If condition (i) in Theorem 1.1.4 is strengthened to everywhere, the following corollary results.
Corollary 1.1.8. Let J be a σ-ideal which contains no nonempty open set. A function f : R → R is continuous everywhere if, and only if, it is J -continuous everywhere.
Proof. If f is continuous, then it is clearly J -continuous. So, suppose f is J -continuous everywhere, x0 ∈ R and ε > 0. Using Proposition 1.1.1(ii), there must be an ordinary open neighborhood G0 of x0 such that F0 = {x ∈ G0 : |f(x) － f(x0)| > ε} ∈ J. Suppose there is an x1 ∈ F0. Choose δ > 0 such that δ < |f(x1) － f(x0)| － ε. As before, there exists an ordinary open neighborhood G1 ⊂ G0 of x1 such that F1 = {x ∈ G1 : |f(x1) － f(x)| > δ} ∈ J. It is clear that G1 ⊂ F0 ∪ F1 ∈ J, because |f(x1) － f(x0)| > ε + δ. But, this implies J contains a nonempty open set, which contradicts the condition placed on J in the statement of the corollary. This contradiction shows that F0 = Φ. The preceding corollary demonstrates that global J -continuity may not be a very useful concept. In particular, it is worthwhile noting for future reference that global I-continuity and global N-continuity are no different than ordinary continuity.
（上記の関連参考：出典URL） http://www.math.wvu.edu/~kcies/ Krzysztof Chris Ciesielski, Ph.D. Professor of Mathematics at Department of Mathematics, West Virginia University and Adjunct Professor at Medical Image Processing Group, Dept. of Radiology, Univ. of Pennsylvania. (抜粋) Books: (with L. Larson and K. Ostaszewski) I-density continuous functions, Memoirs of the AMS vol. 107 no 515, 1994; MR 94f:54035. (引用終り)
http://www.jstor.org/stable/44151978?seq=1#page_scan_tab_contents JOURNAL ARTICLE I-density Continuous Functions Krzysztof Ciesielski, Lee Larson and Krzysztof Ostaszewski Real Analysis Exchange Vol. 15, No. 1 (1989-90), pp. 13-15 Published by: Michigan State University Press （終り）
（参考：用語解説） https://en.wikipedia.org/wiki/Ideal_(set_theory) Ideal (set theory) (抜粋) In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.
More formally, given a set X, an ideal I on X is a nonempty subset of the powerset of X, such that:
1. Φ ∈ I 2.if A∈ I and B⊆ A, then B∈ I, and 3.if A,B∈ I, then A ∪ B∈ I
Some authors add a third condition that X itself is not in I; ideals with this extra property are called proper ideals.
Ideals in the set-theoretic sense are exactly ideals in the order-theoretic sense, where the relevant order is set inclusion. Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.
Contents 1 Terminology 2 Examples of ideals 2.1 General examples 2.2 Ideals on the natural numbers 2.3 Ideals on the real numbers 2.4 Ideals on other sets 3 Operations on ideals 4 Relationships among ideals 5 See also 6 References (引用終り)
https://en.wikipedia.org/wiki/Sigma-ideal σ-ideal Sigma-ideal (Redirected from Σ-ideal) (抜粋) In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra (σ, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.
Let (X,Σ) be a measurable space (meaning Σ is a σ-algebra of subsets of X). A subset N of Σ is a σ-ideal if the following properties are satisfied:
(i) O ∈ N; (ii) When A ∈ N and B ∈ Σ , B ⊆ A ⇒ B ∈ N; (iii) {A_n}_{n∈N }⊆ N→ ∪ _{n∈N }A_n∈ N.
Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. The concept of σ-ideal is dual to that of a countably complete (σ-) filter.
If a measure μ is given on (X,Σ), the set of μ-negligible sets (S ∈ Σ such that μ(S) = 0) is a σ-ideal.
The notion can be generalized to preorders (P,?,0) with a bottom element 0 as follows: I is a σ-ideal of P just when
(i') 0 ∈ I, (ii') x ? y & y ∈ I ⇒ x ∈ I, and (iii') given a family xn ∈ I (n ∈ N), there is y ∈ I such that xn ? y for each n
Thus I contains the bottom element, is downward closed, and is closed under countable suprema (which must exist). It is natural in this context to ask that P itself have countable suprema.
A σ-ideal of a set X is a σ-ideal of the power set of X. That is, when no σ-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the σ-ideal generated by the collection of closed subsets with empty interior. (引用終り)
https://en.wikipedia.org/wiki/Ideal Ideal (抜粋) Mathematics Ideal (ring theory), special subsets of a ring considered in abstract algebra Ideal, special subsets of a semigroup Ideal (order theory), special kind of lower sets of an order Ideal (set theory), a collection of sets regarded as "small" or "negligible" Ideal (Lie algebra), a particular subset in a Lie algebra Ideal point, a boundary point in hyperbolic geometry Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points (引用終り)
（以前のスレから関連抜粋） スレ46 http://2chb.net/r/math/1510442940/398 ＜引用＞ 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 This paper has been published in Gazette of the Australian Mathematical Society, Volume 36, Number 5, November 2009, pp. 353{361. (抜粋) So, in this paper we are going to analyze the dierentiability of the real function fν(x) =0 if x ∈ R ＼ Q, or =1/q^ν if x = p/q ∈ Q, irreducible, for various values of ν ∈ R.
Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not dierentiable) at the rationals, and continuous at the irrationals. With respect the dierentiability, 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 uncountable.
Theorem 2. For ν > 2, let us denote Cν = { x ∈ R : fν is continuous at x }, Dν = { x ∈ R : fν is dierentiable 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)
で、”a nonempty open set”（ordinary open neighborhood ）が、結構重要キーワードじゃないかな？ R中のQのように稠密分散で、 R＼Qは、”a nonempty open set”の集まりになるけれども （似た状況は、上記の「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.」とある通りで） 「422に書いた定理」の系1.8の背理法証明に使えるような、区間(a, b)が取れると言えるかどうかだ？
Thomae（「ポップコーン」）関数の絵が面白いので、ご紹介。 https://arxiv.org/abs/1702.06757 https://arxiv.org/pdf/1702.06757 Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind eta S. Nechaev, K. Polovnikov (Submitted on 22 Feb 2017 (v1), last revised 26 Feb 2017 (this version, v2)) (抜粋) We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers.
The spectral density of ensemble of linear polymer chains distributed with the law ?fL (0<f<1),
where L is the chain length, serves as a particular example.
At f→1, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function.
We suggest a continuous approximation of the popcorn function, based on the Dedekind η-function near the real axis.
Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges.
We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions.
We also pay attention to connection of the Dedekind η-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993. (引用終り)
Research Articles My main research area is Banach space theory but, I have some work in real analysis and know some descriptive set theory as it applies to Banach space theory.
https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf Modifications of Thomae’s function and differentiability, (with James Roberts and Craig Stevenson) Amer. Math. Monthly, 116 (2009), no. 6, 531-535. (抜粋) 3. A DENSE SET. While attempting to prove that T(1/n2) is differentiable on the irrationals, we discovered that quite the opposite is actually true. In fact, as the following proposition indicates, functions that are zero on the irrationals and positive on the rationals will always be non-differentiable on a rather large set.
Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on the irrationals. Then there is an uncountable dense set of irrationals on which f is not differentiable. (引用終り)
Liouville Numbers について、調べていたら、下記ヒット http://www.mathematik.uni -wuerzburg.de/~steuding/elaz2014.pdf On Liouville Numbers - Yet Another Application of Functional Analysis To Number Theory Vortrag auf der ELAZ 2014 in Hildesheim: Jorn Steuding (抜粋) P21/42 Category vs. Measure The set L = (R ＼ Q) ∩ n>=1 (∪q>=2 ∪p (p/q -1/q^n ,p/q+1/q^n )) of Liouville numbers is ・ big in the sense of category (residual, dense Gδ), ・ small in the sense of measure (Lebesgue measure zero, Hausdorff measure zero). For the set of normal numbers it is the other way around. (引用終り)
http://www.mathematik.uni -wuerzburg.de/~steuding/ Prof. Dr. Jorn Steuding Universitat Wurzburg Institut fur Mathematik
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.
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)
>>83 >>R中のQのように稠密分散で、 >>R＼Qは、”a nonempty open set”の集まりになるけれども >？
リウヴィル数をイメージしてもらえば、良いのでは？ 稠密分散で、”a nonempty open set”の集まり 例えば Structure of the set of Liouville numbers より ”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.”
(>>82より再録) "で、”a nonempty open set”（ordinary open neighborhood ）が、結構重要キーワードじゃないかな？ R中のQのように稠密分散で、 R＼Qは、”a nonempty open set”の集まりになるけれども （似た状況は、上記の「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.」とある通りで） 「422に書いた定理」の系1.8の背理法証明に使えるような、区間(a, b)が取れると言えるかどうかだ？"
R＼Qも、リウヴィル数に同じ
つまり、屋上屋の説明だが、RからQを抜く（Qは、孤立点の集合（内点を持たない閉区間の集合）） Rは至る所開（”a nonempty open set”（ordinary open neighborhood ）の集合）
R＼Qの各”a nonempty open set”（ordinary open neighborhood ）は、ここにはq∈Qは含まれない 故に、このような場合には、「422に書いた定理」の系1.8の背理法証明に使えるような、区間(a, b)が取れると言えないのでは？
>>98 >R＼Qも、リウヴィル数に同じ まずリュービル数全体は >Since it is the intersection of countably many such open dense sets のようですので 開集合とは言えませんし実際開集合ではありません 内点を持たないからです 内点を持つなら有理数の稠密性によりリュービル数である有理数がそんざいしてしまいますよ 次に R＼Qですが Qは孤立点の集合ではありません どの有理数の近傍にも必ず有理数が存在するからです また閉集合でもありません 閉包がRだからです ですのでR＼Qもまた開集合にはならないのです
なるほど、”Since it is the intersection of countably many such open dense sets”からは、開集合は言えないのか？ でも、”「422に書いた定理」の系1.8の背理法証明に使えるような、区間(a, b)が取れると言えかどうか”については、どうですか？
http://www.math.ryukoku.ac.jp/~oka/ Welcome to Hiroe's Home Page! Hiroe Oka professor Department of Applied Mathematics and Informatics Faculty of Science and Technology Ryukoku University （龍谷大 岡先生？）
境界の境界 如何なる集合 S についても ∂S ⊇ ∂∂S が成立する。ここで等号は S の境界が内点を持たないとき、かつそのときに限り成り立つ。 これは S が開または閉であるときにも正しい。任意の集合の境界が閉となることから、∂∂S = ∂∂∂S は如何なる集合 S についても成り立つ。したがって、境界をとる操作は弱い意味で冪等である。特に、集合の境界の境界はふつう空でない。
２） 上記とほぼ同じだが、従来のRuler Functionやトマエ関数とその類似の研究で、 ”f(x) = 0 if x is irrational, f(x) = 1/q^2 if x = p/q where p and q are relatively prime integers with q > 0.”(n > 2)
これに匹敵する結果は、>>41-42に書いたが ”Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R. Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite. Then E is co-meager in R (i.e. the complement of a first category set). This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function", Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. ”
つまり、一般な稠密性（但し、H. M. Sengupta and B. K. Lahiriは、可算非可算に関係なく） ”the sets of points at which f is continuous and discontinuous are each dense in R.”なのだが しかし、この discontinuous →リプシッツ連続でないという、上記１）の特性で、定理1.7は拡張されているのだ
