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Statement

Setting (pp. 71--72, Section 7). $R_0=\{0\}$, and for $d\ge1$, $R_d$ consists of $0$ and the numbers $2^{-g_1}+\cdots+2^{-g_t}$ with integers $1\le t\le d$ and $1\le g_1<g_2<\cdots<g_t$: the dyadic numbers in $(0,1)$ with at most $d$ binary digits equal to $1$, together with $0$. The sequence $\omega_0=\{\xi_1,\xi_2,\ldots\}$ is defined by $\xi_1=0$ and, for $k\ge0$,

$$\xi_{2^k+t}=\xi_t+\frac1{2^{k+1}}\qquad(t=1,\ldots,2^k),$$

so $\omega_0=\{0,\frac12,\frac14,\frac34,\frac18,\frac58,\frac38,\frac78,\frac1{16},\ldots\}$; footnote 2 on p. 64 identifies it as Van der Corput's sequence. In this section the sets $S(\kappa)$ are those of the Theorem page, taken for $\omega_0$.

Lemma 6 (p. 72). For every $d\ge1$, the derivative of $R_d$ is $R_{d-1}$.

Lemma 7 (p. 72). For every integer $d\ge0$, $R_d\subseteq S(d)$.

Corollary (p. 72, quoted). "The sets $S^{(d)}(d)$ are non-empty for $d=0,1,2,\cdots$."

By Lemma 6, $R_d^{(d)}=\{0\}$, and by Lemma 7 this lies in $S^{(d)}(d)$. The paper draws the consequence on p. 64: since $S^{(d)}(d)$ is nonempty for $d=1,2,\ldots$, the bound $d>4\kappa$ of the Theorem cannot be replaced by $d>\kappa-\varepsilon$ for any $\varepsilon>0$.

The point $0$ and the term $\xi_1=0$ lie outside $U=(0,1]$, in which the paper's introduction places both the sequence and the sets $S(\kappa)$ (p. 63); Section 7 uses them without comment, and its proof of Lemma 7 starts from "$S(0)$ contains 0" (p. 72). For $d\ge1$ this does not affect the conclusion: removing $0$ from $R_d$ leaves its derived sets unchanged, so $0$ is still a $d$-th order limit point of $S(d)\cap(0,1]$; only the case $d=0$ uses $0\in S(0)$ (an observation of this page). The term $\xi_1=0$ is part of the construction as printed.

Source. W. M. Schmidt, Irregularities of distribution. VI, Compositio Math. 24 (1972), no. 1, 63--74: Section 7 on pp. 71--73, with the remark on sharpness and footnote 2 on p. 64. The edition read is identified on the source card.

Read depth. Claims checked: the construction, Lemmas 6 and 7 and the Corollary were read clause by clause on the printed pages, and the proofs of Lemmas 6 and 7 (pp. 72--73) were read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pages 72--73. Lemma 6 is proved by induction on $d$: a limit of distinct elements of $R_d$ must have its last exponent $g_t$ tending to infinity, so removing the last binary digit gives elements of $R_{d-1}$ with the same limit. Lemma 7 is proved by induction on $d$: the first $2^{g_d}$ terms of $\omega_0$ are the multiples of $2^{-g_d}$ in some order, and by the doubling rule a term falls in $[\hat\eta,\hat\eta+2^{-g_d})$ exactly when its index lies in one residue class modulo $2^{g_d}$, so adding the digit $2^{-g_d}$ to $\hat\eta$ changes the discrepancy by less than $1$.

Dependencies

None outside this section; it sharpens the Theorem (p. 64) by showing its constant is of the right order.

Bears on

• Problem 255: the example shows that a single sequence can have bounded anchored discrepancy at infinitely many points, with derived sets of every finite order; it does not bear on whether some interval has unbounded discrepancy, which the Corollary on p. 64 addresses.