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Statement

Notation as on the Theorem page: for an arbitrary sequence $\xi_1,\xi_2,\ldots$ in $U=(0,1]$, $D(n,\alpha)=\lvert Z(n,\alpha)-n\alpha\rvert$ with $Z(n,\alpha)$ the number of $i\le n$ with $0\le\xi_i<\alpha$; $S(\kappa)$ is the set of $\alpha\in U$ with $D(n,\alpha)\le\kappa$ for all $n\ge1$, and $S(\infty)$ is the union of the $S(\kappa)$.

Corollary (p. 64, quoted). "The sets $S(\kappa)$ are at most countable and they are nowhere dense. The set $S(\infty)$ is at most countable."

The paper obtains it from the Theorem through the remark, made just before it on p. 64, that induction on $d$ shows a set of reals with empty $d$-th derived set to be at most countable and nowhere dense. Since $S(\kappa)\subseteq S(\lceil\kappa\rceil)$, $S(\infty)$ is the union of the countably many sets $S(0),S(1),S(2),\ldots$ (an observation of this page).

The paper sets the Corollary against earlier work (p. 63): in part I of the series (Quart. J. Math. Oxford (2) 19 (1968), 181--191) Schmidt answered Erdős's question whether $S(\infty)$ must be a proper subset of $U$, showing among other things that $S(\infty)$ has Lebesgue measure zero. For the sequence of fractional parts $\{\theta\},\{2\theta\},\ldots$ with $\theta$ irrational, it recalls (p. 64) that the numbers $\{k\theta\}$, $k$ an integer, belong to $S(\infty)$ (Hecke) and are its only elements (Kesten, answering a question of Erdős and Szüsz), so that there $S(\infty)$ is countable. Intervals $\alpha<\xi\le\beta$ not anchored at $0$ behave differently: for that sequence the discrepancy of such an interval is bounded if (Ostrowski) and only if (Kesten) its length is some $\{k\theta\}$, so continuum many intervals have bounded discrepancy (p. 64).

Source. W. M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), no. 1, 63--74: the setting on p. 63, the Corollary and the background on p. 64. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the deduction from the Theorem were read clause by clause on the printed page; the inductive remark on derived sets is standard and was not written out here. Nothing here is independently reviewed.

Proof pointer

Page 64. Apply the Theorem with any integer $d>4\kappa$: $S(\kappa)$ has empty $d$-th derived set, and a set of reals with empty $d$-th derived set is at most countable and nowhere dense, by induction on $d$.

Dependencies

Theorem (p. 64) of the same paper.

Bears on

• Problem 255: the problem asks whether every sequence $z_1,z_2,\ldots$ in $[0,1]$ has an interval $I\subseteq[0,1]$ with $\limsup_N\lvert D_N(I)\rvert=\infty$, where $D_N(I)=\#\{n\le N:z_n\in I\}-N\lvert I\rvert$. For $I=[0,\alpha)$ with $\alpha\in(0,1]$, $\lvert D_N(I)\rvert$ is Schmidt's $D(N,\alpha)$, and $\alpha\notin S(\infty)$ means exactly that this limsup is infinite. For a sequence in $(0,1]$, the Corollary therefore gives the interval $[0,\alpha)$ for every $\alpha$ in $(0,1]$ outside a countable set. The Corollary is stated for sequences in $(0,1]$; the problem also admits terms equal to $0$, a case the printed statement does not cover, although the paper's own Section 7 example begins with the term $0$. The problem's claim page for this paper records the paper's claim on the problem.