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Statement

Pp. 182--183 define, for each integer $n\ge0$, the unique sequence $\varepsilon(n)=(\varepsilon_1(n),\varepsilon_2(n),\ldots)$ with

• (i) $n=\sum_{i\ge1}\varepsilon_i(n)F_{2i}$;
• (ii) $\varepsilon_i(n)\in\{0,1,2\}$ for every $i$;
• (iii) between any two digits equal to $2$ lies a digit $0$: if $\varepsilon_i(n)=\varepsilon_j(n)=2$ with $i<j$, then $\varepsilon_k(n)=0$ for some $k$ with $i<k<j$

(existence and uniqueness are Lemma 1, p. 185), and the sequence $\bar x^*=(x^*_0,x^*_1,\ldots)$ by

$$x^*_n=\alpha\sum_{i\ge1}\frac{\varepsilon_i(n)}{F_{2i}}, \qquad \alpha=\Bigl(1+\sum_{k\ge1}\frac1{F_{2k}}\Bigr)^{-1},$$

noting that $x^*_n\in[0,1]$ and that $\bar x^*$ is nowhere dense. As printed on p. 183:

Theorem 2.

$$C(\bar x^*)=\alpha. \tag{3}$$

In fact,

$$\inf_{n\ge1}\ \inf_{m\ge0}\ n\,|x^*_{m+n}-x^*_m|=\alpha.$$

Here $C(\bar x)=\inf_n\liminf_{m\to\infty}n|x_{m+n}-x_m|$ (p. 182). With Theorem 1 this shows that $\alpha$ is the largest constant for which Theorem 1 holds.

Source. F. R. K. Chung and R. L. Graham, On irregularities of distribution, Finite and Infinite Sets (Eger, 1981), Colloq. Math. Soc. János Bolyai 37, North-Holland (1984), 181--222; the definitions on printed pp. 182--183 and Theorem 2 on p. 183 (PDF pp. 2--3 of the image-only file), read on the rendered page images; the extremal-sequence section on pp. 212--219 (PDF pp. 32--39). The edition is identified in the source digest.

Read depth. Claims checked: the definition of $\varepsilon(n)$ and $\bar x^*$ and the two displays of Theorem 2 were read clause by clause on the page image; the proof was read for its structure only.

Proof pointer

The section "An extremal sequence" (pp. 212--219) defines $y(n)=\sum_{i\ge1}\varepsilon_i/F_{2i}$ from the representation of Lemma 1 and proves the Theorem (35): $|(a-b)(y(a)-y(b))|\ge1$ for all $a\ne b$, by cases on the digit strings (the case $b=0$ reduces to $y(a)\ge1/a$, checked at $a=F_{2m}$; otherwise one may assume $\varepsilon^{(a)}_i\varepsilon^{(b)}_i=0$ for all $i$ and split according to whether $a$ (Case 1, pp. 212--214) or $b$ (Case 2, pp. 214--219) carries the lowest-index nonzero digit). Since $x^*_n=\alpha y(n)$, (35) gives $n|x^*_{m+n}-x^*_m|\ge\alpha$ for all $m\ge0$, $n\ge1$, which with Theorem 1 yields the displayed equalities. Not reconstructed here. The concluding remarks (p. 220) add that $x'_n=\{n\tau\}$, $\tau=\frac12(1+\sqrt5)$, has $C(\bar x')=\frac{3-\sqrt5}2=0.381966\ldots<\alpha$, although its first $n$ terms are always order isomorphic to those of $\bar x^*$.

Dependencies

Lemma 1 (p. 185) for the representation; Lemma 2 (p. 186) and Lemma 3 (p. 188), which the proof of (35) cites on pp. 214--219; Theorem 1 for the upper bound; the Fibonacci identities of pp. 184--185.

Bears on

• Problem 480: the site's commentary says the authors "also prove that this constant is best possible"; this is that statement, and the site's discussion thread describes the same construction (the digits $e_k(n)$ with the 0-between-two-2s rule).