Khintchine 1923 ein satz uber kettenbruche mit

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problem_p303: Khintchine's question whether, for a fixed Lebesgue measurable set E in (0,1), the multiples kx visit E with asymptotic frequency mE for all x outside a set of measure zero.

theorem_1: Khintchine's theorem that for every positive function φ(n) tending to 0 some irrational x fails the relation sum_{k<=n} ρ(kx) - n/2 = O(nφ(n)), where ρ is the fractional part.

theorem_2: Khintchine's theorem that for every ε > 0 and every x outside a set of measure zero, the sum of the fractional parts ρ(kx) over k <= n differs from n/2 by o(lg^{1+ε} n).

theorem_3: Khintchine's theorem that for every x outside a set of measure zero the absolute difference between the sum of ρ(kx) over k <= n and n/2 is Ω(lg n), Ω being the negation of O.

theorem_p289: Khintchine's main theorem: for every ε > 0 and every x outside a set of Lebesgue measure zero, the sum A_n(x) of the first n partial quotients of the regular continued fraction of x is o(n^{1+ε}).

theorem_p298: Khintchine's theorem that for an interval δ in (0,1), every ε > 0 and every x outside a set of measure zero, the number F(n, δ, x) of the points ρ(x), ..., ρ(nx) lying in δ is δn + o(lg^{1+ε} n).

theorem_p302: Khintchine's theorem that for a closed polygon P dilated by t from a fixed origin, for almost every direction α of the axes, the lattice-point count of P_t differs from its area by O(lg^{1+ε} t) for every ε > 0.

theorem_p305: Khintchine's theorem that a system E of infinitely many pairwise disjoint intervals in (0,1) satisfies (6) and (7) for almost every x whenever its tail sums R_n are O(1/n^ε) for some ε > 0, in particular when its lengths are O(1/n^{1+ε}).

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Khintchine, A., Ein Satz über Kettenbrüche, mit arithmetischen Anwendungen. Math. Z. 18 (1923), 289--306.

This German-language paper (a Göttingen digitization; the first page is the library's cover sheet, and the article runs pp. 289--306) proves in § 1 its main theorem: for every $\varepsilon>0$ and every $x$ outside a set of Lebesgue measure zero, the sum $A_n(x)$ of the first $n$ partial quotients of the regular continued fraction of $x$ is $o(n^{1+\varepsilon})$ (p. 289), a complement to a theorem of F. Bernstein. § 2 applies it to Sierpiński's theorem $\sum_{k\le n}\rho(kx)-\frac n2=o(n)$, $\rho$ the fractional part: no faster rate $O(n\varphi(n))$ with $\varphi\to0$ holds for every irrational $x$ (Satz 1, pp. 291--292), but for every $\varepsilon>0$ and almost every $x$ the difference is $o(\lg^{1+\varepsilon}n)$ (Satz 2, p. 293), and almost everywhere it is $\Omega(\lg n)$, $\Omega$ the negation of $O$ (Satz 3, p. 296). § 3 proves the same almost-everywhere bound $F(n,\delta,x)-\delta n=o(\lg^{1+\varepsilon}n)$ for the number $F(n,\delta,x)$ of $\rho(kx)$, $k\le n$, in an interval $\delta\subseteq(0,1)$ (p. 298), contrasting it with Hardy and Littlewood's almost-everywhere bounds $O(\sqrt{n\lg n})$ and $\Omega(\sqrt n)$ for the sequence $\rho(a^kx)$. § 4 applies §§ 2--3 to lattice points: for a closed polygon $P$ dilated by $t$, for almost every direction of the axes the lattice-point count minus the area is $O(\lg^{1+\varepsilon}t)$ for every $\varepsilon>0$ (p. 302).

§ 5, "Ein neues Problem" (pp. 303--306), notes that for a Jordan measurable $E\subseteq(0,1)$ with periodic characteristic function $g$, every irrational $x$ satisfies (6) $\sum_{k\le n}g(kx)-n\,mE=o(n)$, equivalently (7) the averages tend to $mE$, while for Lebesgue measurable $E$ this fails in general for some irrational $x$; Khintchine asks whether (6) and (7) then hold for all $x$ outside a set of measure zero (p. 304). He reduces the question to countable unions of pairwise disjoint intervals, says this case still seems difficult, and proves with a measure lemma (Hilfssatz, p. 304) and the result of § 3 that such a union is regular, that is satisfies (6) and (7) almost everywhere, whenever its tail sums of lengths are $O(n^{-\varepsilon})$ for some $\varepsilon>0$ (Satz, p. 305).

Source: https://gdz.sub.uni-goettingen.de/id/PPN266833020_0018. The digitization's first page is the digitizer's terms sheet, which prints "The Goettingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes ... Some of our collections are protected by copyright. Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen State- and University Library.", not the publisher's line, every other right reserved.

Bears on. #994: the question of § 5 (p. 304) is the problem's question, posed by Khintchine for a fixed Lebesgue measurable $E\subseteq(0,1)$ and almost all $x$; the paper answers it yes only for unions of disjoint intervals with tail sums $O(n^{-\varepsilon})$ (Satz, p. 305) and proves no general answer.

Read status: claims checked for the theorems listed below, the Hilfssätze of pp. 300--301 and p. 304 and the question of § 5, read clause by clause on the page images of the print; the proofs were followed for structure. Nothing here is independently reviewed.

Results.

• Satz (§ 1, p. 289): for every $\varepsilon>0$, $A_n(x)=o(n^{1+\varepsilon})$ for almost all $x$.
• Satz 1 (pp. 291--292): for every positive $\varphi(n)\to0$ some irrational $x$ fails $\sum_{k\le n}\rho(kx)-\frac n2=O(n\varphi(n))$.
• Satz 2 (p. 293): for every $\varepsilon>0$ and almost all $x$, $\sum_{k\le n}\rho(kx)-\frac n2=o(\lg^{1+\varepsilon}n)$.
• Satz 3 (p. 296): for almost all $x$, $\bigl|\sum_{k\le n}\rho(kx)-\frac n2\bigr|=\Omega(\lg n)$.
• Satz (§ 3, p. 298): for every $\varepsilon>0$ and almost all $x$, $F(n,\delta,x)-\delta n=o(\lg^{1+\varepsilon}n)$.
• Satz (§ 4, p. 302), with its Hilfssatz (pp. 300--301): lattice points in dilated polygons for almost every direction of the axes.
• Problem (§ 5, pp. 303--304): for a Lebesgue measurable $E$, do (6) and (7) hold for almost all $x$?
• Satz (§ 5, p. 305), with its Hilfssatz (p. 304): a disjoint union of intervals with $R_n=O(n^{-\varepsilon})$ for some $\varepsilon>0$ is regular.

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