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Statement

Let $\alpha=(1+\sum_{k\ge1}F_{2k}^{-1})^{-1}=0.39441967\ldots$ be the constant of Theorem 1, with $F_0=0$, $F_1=1$, $F_{n+2}=F_{n+1}+F_n$. For each integer $n\ge0$ the announcement takes the unique sequence $\varepsilon(n)=(\varepsilon_1(n),\varepsilon_2(n),\ldots)$ such that

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

and defines $\bar x^*=(x^*_0,x^*_1,\ldots)$ by

$$x^*_n=\alpha\sum_{i\ge1}\varepsilon_i(n)F_{2i}^{-1}.$$

The page notes that each $x^*_n$ lies in $[0,1]$ and that $\bar x^*$ is nowhere dense; it asserts the uniqueness in the definition without proof.

Theorem 2 (p. 4001). With $C(\bar x)=\inf_n\liminf_{m\to\infty}n|x_{m+n}-x_m|$,

$$C(\bar x^*)=\alpha, \tag{2}$$

and in fact

$$\inf_n\inf_m n\,|x^*_{m+n}-x^*_m|=\alpha. \tag{3}$$

The print writes the infima in (3) without ranges; the sequence is indexed from $x^*_0$, so $m$ runs over $m\ge0$ and $n$ over $n\ge1$, the ranges the 1984 chapter prints. With Theorem 1, (2) shows that the constant $\alpha$ in Theorem 1 cannot be replaced by any smaller constant.

Source. F. R. K. Chung and R. L. Graham, On irregularities of distribution of real sequences, Proc. Natl. Acad. Sci. USA 78 (1981), no. 7, 4001; the definition of $\varepsilon(n)$ and $\bar x^*$ and Theorem 2 are on the one printed page. The edition is identified in the source digest.

Read depth. Claims checked: the conditions (i)--(iii), the definition of $\bar x^*$ and the displays [2] and [3] were read clause by clause on the page image. The page gives no proof.

Proof pointer

None on the page. The proof is Theorem 2 of the 1984 chapter (p. 183 there, proved in its section on an extremal sequence, pp. 212--219).

Dependencies

Theorem 1 for the upper bound $C(\bar x)\le\alpha$, which with (3) gives (2); the existence and uniqueness of the representation $\varepsilon(n)$, asserted on the page and proved as Lemma 1 of the 1984 chapter.

Bears on

• Problem 480: the announced statement that the constant $\alpha$ in the bound $C(\bar x)\le\alpha$ of Theorem 1 is best possible: with Theorem 1, $\alpha$ is the least constant $c$ with $C(\bar x)\le c$ for every sequence in $[0,1]$, below the problem's $5^{-1/2}$. The theorem is stated here without proof.