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Statement

Satz (§ 4, p. 302). Let $P$ be an arbitrary closed polygon in a plane, and let the origin $O$ of coordinates be chosen arbitrarily but fixed. The position of the rectangular axes $Ox$, $Oy$ is determined by the direction coefficient $\alpha$ of $Ox$ relative to a fixed direction. Let $P_t$ be the polygon $P$ dilated from $O$ in the ratio $1:t$. Then, as $t$ grows, for all values of $\alpha$ outside a set of measure zero,

$$B(P_t)-J(P_t)=O(\lg^{1+\varepsilon}t)$$

for every $\varepsilon>0$, where $J$ is area and $B$ is the number of lattice points inside.

The quantifiers are as printed: the exceptional set of $\alpha$ is named before "for every $\varepsilon>0$".

Hilfssatz (pp. 300--301). Let $0<a<b$ with $a\equiv b\equiv\frac12\pmod 1$, $\theta$ constant and $m$ arbitrary, and let $G$ be the trapezoid bounded by the lines $y=\frac12$, $x=a$, $x=b$ and $y=\theta x+m$ in rectangular coordinates. (The print lists the third line as $y=b$; Fig. 1 and the proof show the vertical line $x=b$.) Then, with $a$, $b$, $m$ variable, for every $\varepsilon>0$ and every $\theta$ outside a set of measure zero (at most), $B(G)-J(G)=O(\lg^{1+\varepsilon}b)$.

Proof pointer

Hilfssatz, pp. 301--302: write $B(G)$ as a sum of integer parts $[\theta k+m]$ over $a\le k\le b$ and estimate the fractional-part sums with Satz 2 and the count of $k$ with $\rho(\theta k)\ge1-\rho(m)$ with the Satz of § 3. Satz, pp. 302--303: the lines $x=\pm\frac12$, $y=\pm\frac12$ cut $P_t$ into nine parts; the five inside the cross contribute $O(1)$, and each remaining part is a signed sum of a $t$-independent number of trapezoids of the Hilfssatz's form, up to $O(1)$; footnote 10 notes that all values of $\alpha$ outside a null set make every slope $\theta_i$ admissible.

Read depth

Claims checked: the Hilfssatz and the Satz were read clause by clause on the page images of the print, and the proofs were followed for structure. Nothing here is independently reviewed.

Dependencies

Satz 2 (p. 293) and 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

No Erdős problem directly.