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Statement

Setting (§ 3, p. 297). $\delta=(\alpha,\beta)$ is an interval contained in $(0,1)$, and $\delta$ also denotes its length. $g$ is the characteristic function of $\delta$, extended to the real line with period $1$, and

$$F(n,\delta,x)=\sum_{k=1}^{n}g(kx),$$

the number of the points $\rho(x),\rho(2x),\ldots,\rho(nx)$ (the paper's sequence (3)) that lie in $\delta$.

Satz (§ 3, p. 298). For every $\varepsilon>0$ and every $x$ outside a set of measure zero (at most),

$$F(n,\delta,x)-\delta n=o(\lg^{1+\varepsilon}n).$$

Context given by the paper (pp. 297--298). For every irrational $x$, $F(n,\delta,x)-\delta n=o(n)$; the author says he does not know who first proved it. A sharper estimate for all irrationals is impossible, as one sees by adapting the proof of Satz 1. Hardy and Littlewood (Acta Math. 37 (1914)) studied the sequence $\rho(a^kx)$, $k=0,\ldots,n-1$, for a natural number $a$, and got $F(n,\delta,x)-\delta n=O(\sqrt{n\lg n})$ and $\Omega(\sqrt n)$ for all $x$ outside a null set; footnote 8 adds that both estimates also hold for the sequence $(1!x),(2!x),\ldots,(n!x)$. On p. 300 the paper says the Satz can probably be sharpened but the order of the remainder cannot be pushed down to $\lg n$, these questions being handled as in § 2.

Proof pointer

Pp. 298--300. With $q_i\le n<q_{i+1}$ and $n=S_0q_i+R_0$ as in Satz 2, counting the fractions $\nu/q_i$ in $\delta$ gives

$$|F(n,\delta,x)-\delta n| <\Bigl|\sum_{k=1}^{R_0}g(kx)-\delta R_0\Bigr|+2a_{i+1}(x)+4$$

(p. 299), and the rest runs as the end of the proof of Satz 2, with the Satz of § 1.

Read depth

Claims checked: the setting, the statement and the surrounding remarks were read clause by clause on the page images of the print, and the proof was followed for structure. Nothing here is independently reviewed.

Dependencies

Satz of § 1 (p. 289) and the argument of Satz 2 (pp. 293--295).

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: the Satz is the case of a single interval $E=\delta$, with an error term far smaller than $o(n)$; the paper uses it to prove the Satz of § 5 for unions of intervals. For a single interval the problem's relation already holds for every irrational $x$, as the paper recalls.