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Statement

Setting (§ 5, p. 305). $E$ is a system of infinitely many intervals in $(0,1)$, pairwise without common points, listed in some fixed order $\delta_1,\delta_2,\ldots$, where $\delta_i$ also denotes the length. The paper sets $S_n=\sum_{i=1}^{n}\delta_i$ and $R_n$ the tail sum of the lengths; the print's lower limit of summation for $R_n$ reads $i=n-1$, while the proof uses $S_{A(n)}$ and $R_{A(n)}$ as complementary parts of $E$, which fits $i=n+1$. The hypothesis below is the same under either reading. The system $E$ is called regular when (6) and (7) of the problem of § 5 hold for every $x$ outside a set of measure zero (at most).

Satz (p. 305). The system $E$ is regular whenever there is an $\varepsilon>0$ with

$$R_n=O\Bigl(\frac1{n^{\varepsilon}}\Bigr).$$

In particular $E$ is regular when $\delta_n=O(1/n^{1+\varepsilon})$ for some $\varepsilon>0$.

Hilfssatz (p. 304). Let $E_1,E_2,\ldots$ be Lebesgue measurable sets with characteristic functions $g_1,g_2,\ldots$, so that $k_n(x)=\sum_{i=1}^{n}g_i(x)$ counts the sets among $E_1,\ldots,E_n$ containing $x$. Let $\psi(n)$ be a function of the integer argument $n$ with $\psi(n+1)>\psi(n)>0$ and $\sum_{i=1}^{\infty}mE_i/\psi(i)$ convergent. Then $k_n(x)=O(\psi(n))$ for every $x$ outside a set of measure zero (at most).

Proof pointer

Hilfssatz, pp. 304--305: integrate $\sum_i g_i(x)/\psi(i)$ term by term; the series converges almost everywhere, and $k_n(x)\le\psi(n)\sum_{i\le n} g_i(x)/\psi(i)$ since $\psi$ increases. Satz, pp. 305--306: put $A(n)=[n/\lg^2n]$. By the Satz of § 3, almost every $x$ has $|F(n,\delta_i,x)-\delta_in|<B\lg^{3/2}n$ with $B$ independent of $n$ and $i$, which handles the first $A(n)$ intervals, (8). The remaining part $R_{A(n)}$ is handled by the Hilfssatz with $\psi(k)=k^{1-\varepsilon/2}$, (9), and together these give $F(n,E,x)-n\,mE=o(n)$.

Read depth

Claims checked: the Hilfssatz, the definition of a regular system and the Satz were read clause by clause on the page images of the print, and the proofs were followed for structure. The uniformity in $i$ claimed for the bound in (8) was not checked. Nothing here is independently reviewed.

Dependencies

Satz of § 3 (p. 298).

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 answers the problem's question yes for every set $E$ that is a union of infinitely many pairwise disjoint intervals in $(0,1)$ whose tail sums of lengths are $O(n^{-\varepsilon})$ for some $\varepsilon>0$ in some ordering. It says nothing about other measurable sets.