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Statement

Notation as on the Satz I page; $N(J,x)$ counts the points $R(k\alpha)$, $1\le k\le x$, in the interval $J$ of length $(J)$, intervals being taken modulo $1$.

Equation (4) (p. 36). If

$$\Bigl|\alpha-\frac pq\Bigr|\ge\frac{c}{q^{r+1}},\qquad r>1,\quad c>0,$$

for all integers $q\ge1$, then

$$N(J,x)-(J)x=O\bigl(x^{1-\frac1r}\bigr) \tag{4}$$

for arbitrary intervals $J$. A footnote (p. 36) says that Hecke first proved $N(J,x)-(J)x=O(x^a)$, for suitable $a$ with $0<a<1$, uniformly for all intervals of length $(J)$, and that the exponent $1-\frac1r$ came only from the author's own arguments. The paper's aim here is a new, much shorter derivation of (4) that rests on Satz II and is arranged to extend to several dimensions (p. 36).

The function $\nu(t)$ (p. 41). Let $\alpha$ be real, rational allowed. For each $t>1$ let $\nu(t)$ be such that $R(k\alpha)>\frac1t$ and $1-R(k\alpha)>\frac1t$ for all integers $k$ with $0<k<\nu(t)$, with $\nu(t)$ nondecreasing in $t$. Dirichlet's theorem then gives $\nu(t)\le t$, and for rational $\alpha$, $\nu(t)$ stays below the denominator of $\alpha$ (pp. 41--42).

The maximal discrepancy and (13) (pp. 43--44). For $x\ge1$ let $A(x)$ be the supremum of $|N(k,J)-(J)k|$ over all integers $k$ with $0<k\le x$ and all intervals $J$ that can be taken modulo $1$ as subintervals of $[0,1)$; plainly $A(x)\le x$. Then

$$A(x)\le\frac{2x}{\nu(t)}+2A(t), \tag{13}$$

where $t>1$ is left free in the derivation (p. 42). A footnote there adds the condition, as printed, $\nu(t)>\frac\lambda2$, with $\lambda=(J)$. The construction uses an interval of length $\lambda-\frac2{\nu(t)}$, which needs $\nu(t)>\frac2\lambda$, so the printed inequality appears to be a misprint for that one (an observation of this page). A footnote on p. 44 says that intervals of the kind in Satz I give the sharper form $A(x)\le\frac{2x}{\nu(t)}+A(t)$ (13'), compare Hecke's formula (22).

Equation (14) (p. 44). If $\nu(t)=ct^{1/r}$ with $0<c\le1$ and $r>1$, then for all $x\ge1$

$$A(x)<\frac{4^{\frac{r}{r-1}}}{c}\,x^{1-\frac1r}. \tag{14}$$

The paper notes that such a $\nu(t)$ can be taken for every algebraic number, with suitable $r$. From (13') the same route gives the constant $2^{\frac{2r-1}{r-1}}/c$ in place of $4^{\frac{r}{r-1}}/c$ (14', p. 45). If $\alpha$ is rational and $\nu(t)=ct^{1/r}$ can be taken for all $t\le T$, then (14) holds for all $x\le4^{\frac{r}{r-1}}T$ (p. 45).

From (4)'s hypothesis to (14) (an observation of this page; the paper does not spell it out). If $0<k$ and $R(k\alpha)\le\frac1t$ or $1-R(k\alpha)\le\frac1t$, then $|k\alpha-p|\le\frac1t$ for some integer $p$, and the hypothesis of (4) gives $|k\alpha-p|\ge ck^{-r}$, so $k\ge(ct)^{1/r}$. Hence $\nu(t)=(ct)^{1/r}$ is admissible, and with $c\le1$ (which can be assumed) this is of the form required in (14).

The case $r=1$ (pp. 45--46). The paper says this proof of $A(x)=O(x^{1-1/r})$ for $r>1$ is much simpler than the two known ones, while its first, continued-fraction proof also shows that the order of magnitude of the estimate, which the print here numbers (15), is "die richtige" (p. 45). It says this route apparently no longer gives the sharpest bound $A(x)=O(\lg x)$ (16) for $r=1$, also the right one, so that its continued-fraction proof of (16), in the 1922 paper cited on p. 36, remains for now the only one. A footnote (p. 45) notes that for irrational $\alpha$, $r=1$ exactly when $\alpha$ has bounded continued-fraction partial quotients. From (13) with $\nu(t)=ct$, $0<c\le1$, the paper derives instead

$$cA(x)<2e^{2\sqrt{\lg2}\sqrt{\lg x}}\quad(x\ge1), \tag{19}$$

so $A(x)=O\bigl(e^{2\sqrt{\lg2\,\lg x}}\bigr)$ (17) (p. 45), and from (13') the bound $cA(x)<2\sqrt2\,e^{\sqrt{2\lg2}\sqrt{\lg x}}$ (19'), so $A(x)=O\bigl(e^{\sqrt{2\lg2\,\lg x}}\bigr)$ (17') (p. 46).

Source. Alexander Ostrowski, Mathematische Miszellen. XVI. Zur Theorie der linearen Diophantischen Approximationen, Jber. Deutsch. Math.-Verein. 39 (1930), 34--46; (4) on p. 36, Section IV with $\nu(t)$ and (8)--(12) on pp. 41--43, $A(x)$ and (13) on pp. 43--44, Section V with (14) on p. 44 and (14'), (16), (17) on pp. 45--46. The edition read is identified on the source card.

Read depth. Claims checked: (4), the definitions of $\nu(t)$ and $A(x)$, and (13), (13'), (14), (14'), (17), (17'), (19), (19') were read on the page images. The derivations were followed but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 41--45. For any real $\varrho$ some integer $x$ with $0<x\le t$ has $x\alpha$ within $\frac1{\nu(t)}$ of $\varrho$ modulo $1$ ((9), p. 42). Given $J$ of length $\lambda$, Satz II supplies intervals $J_\pm$ of lengths $\lambda\pm\frac2{\nu(t)}$ whose counts are at most, respectively at least, $x\lambda_\pm$. Shifting by a $q_\pm\le t$ chosen through (9) places $J$ inside $J_+$, and $J_-$ inside $J$, for the tail of the orbit, which gives the one-sided bounds (11) and (12) with errors controlled by $A(t)$ (p. 43); together they give (13). Then (14) follows by induction on $x$: it holds trivially for $1\le x<4^{r/(r-1)}/c$, since $A(x)\le x$; an $x$ at which (14) fails while it holds at $t=x/4^{r/(r-1)}$ is contradicted by (13) at that $t$ (p. 44).

Dependencies

Satz II (p. 36) and Dirichlet's approximation theorem ((8), p. 41).

Bears on

None recorded. The estimate bounds the growth of the discrepancy of intervals of every length; it gives no bounded discrepancy and so does not bear on Problem 998, which asks which intervals have bounded discrepancy.