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Statement

Setting (§ 5, "Ein neues Problem", p. 303). $E$ is a point set in the interval $(0,1)$ and $g$ its characteristic function, extended to the real line with period $1$; $mE$ is the Lebesgue measure of $E$. The relations are

$$\text{(6)}\quad \sum_{k=1}^{n}g(kx)-n\,mE=o(n), \qquad \text{(7)}\quad \lim_{n\to\infty}\frac1n\sum_{k=1}^{n}g(kx)=mE,$$

which the paper treats as saying the same thing.

What the paper records (pp. 303--304). When $E$ is Jordan measurable, one infers easily from the interval case that (6) and (7) hold for every irrational $x$. When $E$ is only Lebesgue measurable, (6) and (7) in general no longer hold for every irrational $x$, which the paper says is easily seen.

Problem (p. 304, quoted). "Es entsteht nun die für die Funktionentheorie sehr wichtige Frage, ob nicht in diesem Falle die Beziehungen (6), (7) für alle $x$ mit Ausnahme höchstens einer Menge vom Maße Null gelten."

In the corpus's words: for a fixed Lebesgue measurable $E\subseteq(0,1)$, do (6) and (7) hold for every $x$ outside a set of measure zero? The set $E$ is fixed before the exceptional null set, which may depend on it.

The paper adds (p. 304) that an affirmative answer would give a definition of measure and integral of arithmetic nature, that some convergence questions for Fourier series seem connected with it, and that the problem reduces easily to the case where $E$ is the union of countably many intervals with no common points, a case that still seems to present many difficulties. It then answers the question affirmatively for a large class of such unions in the Satz of § 5.

Read depth

Claims checked: the setting, (6), (7) and the question were read clause by clause on the page images of the print. The paper proves no general answer. Nothing here is independently reviewed.

Dependencies

None.

Source. A. Khintchine, Ein Satz über Kettenbrüche, mit arithmetischen Anwendungen, Math. Z. 18 (1923), 289--306; the edition read is named on the source card.

Bears on

• Problem 994: this question is the problem in the form of its precise statement (the set $E$ fixed first, then almost all $x$), posed here by Khintchine; the paper answers it only for the special class of the Satz of § 5.