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Statement

Setting (p. 63). $U$ is the interval $0<\xi\le1$ and $\omega=\{\xi_1,\xi_2,\ldots\}$ is an arbitrary sequence in $U$. For $\alpha\in U$ and a positive integer $n$, $Z(n,\alpha)$ is the number of indices $i$ with $1\le i\le n$ and $0\le\xi_i<\alpha$, and

$$D(n,\alpha)=\lvert Z(n,\alpha)-n\alpha\rvert .$$

For $\kappa\ge0$, $S(\kappa)$ is the set of $\alpha\in U$ with $D(n,\alpha)\le\kappa$ for every $n=1,2,\ldots$; $S(\infty)$ is the union of the sets $S(\kappa)$, the $\alpha\in U$ at which $D(n,\alpha)$ stays bounded in $n$.

Derived sets (pp. 63--64). A number $\gamma$ is a limit point of a set $S$ when some sequence of distinct elements of $S$ converges to $\gamma$; the derivative $S^{(1)}$ is the set of limit points of $S$, and $S^{(d)}=(S^{(d-1)})^{(1)}$ for $d=2,3,\ldots$.

Theorem (p. 64, quoted). "Suppose $d>4\kappa$. Then $S^{(d)}(\kappa)$ is empty."

Here $d$ is a positive integer and $\kappa\ge0$, and $S^{(d)}(\kappa)$ is the $d$-th derived set of $S(\kappa)$. The sequence is arbitrary: no uniform distribution or other hypothesis is placed on it.

The paper remarks (p. 64) that the quantity $4\kappa$ could be somewhat reduced at the cost of further complications, and that the example of Section 7 shows it cannot be replaced by $\kappa-\varepsilon$ for any $\varepsilon>0$; see the Corollary on p. 72.

Source. W. M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), no. 1, 63--74: the setting on p. 63, the Theorem on p. 64, the reduction to the Proposition in Section 2 (pp. 64--65), the proof of the Proposition in Sections 3--6 (pp. 65--71). The edition read is identified on the source card.

Read depth. Claims checked: the setting, the definitions and the statement were read clause by clause on the printed pages, and the deduction of the Theorem from the Proposition (p. 65) was followed. The proof of the Proposition (pp. 65--71) was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 64--71. With $f(n,\alpha)=Z(n,\alpha)-n\alpha$, Section 2 measures the oscillation $h(I,\alpha)$ of $f(\cdot,\alpha)$ over an interval $I$ of consecutive integers, the maximum minus the minimum. Its Proposition (p. 65) says that if the $d$-th derived set of a set $R$ meets $(0,1)$, then for every $\varepsilon>0$ there are $2^d$ elements of $R$, with neighbourhoods, such that the average of the oscillations at any points of those neighbourhoods exceeds $\frac12(d+1)-\varepsilon$ on every long enough interval. Some element of $R$ then has oscillation above that value, so its discrepancy exceeds $\frac14(d+1)-\frac12\varepsilon$ at some $n$; taking $R=S(\frac14(d+1)-\varepsilon)$ gives a contradiction, so that derived set lies in $\{0,1\}$ and the next one is empty, which yields the Theorem (p. 65). The Proposition is proved by induction on $d$: the case $d=0$ is Lemma 1 (p. 65), derived from Lemma 2 (p. 66) through Kronecker's theorem; Lemmas 3 and 4 (Section 4, pp. 66--69) vary Lemma 2 for pairs of points; Lemma 5 (Section 5, pp. 69--70) is an inequality combining oscillations over subintervals; Section 6 (pp. 70--71) carries out the induction.

Dependencies

The Proposition (p. 65) and Lemmas 1--5 of the same paper; Kronecker's theorem on inhomogeneous approximation.

Bears on

• Problem 255: the Theorem is the source of the countability statement of the Corollary on p. 64, whose page states the relation to the problem.