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Statement

For a sequence $\bar x=(x_1,x_2,\ldots)$ of real numbers with $x_k\in[0,1]$ the chapter defines (p. 182)

$$C(\bar x)\equiv\inf_n\liminf_{m\to\infty}n\,|x_{m+n}-x_m|,$$

the infimum over positive integers $n$. As printed on p. 182:

Theorem 1. "For any sequence $\bar x$ in $[0,1]$,

$$C(\bar x)\le\Bigl(1+\sum_{k\ge1}\frac1{F_{2k}}\Bigr)^{-1}\equiv\alpha=0.39441967\ldots, \tag{2}$$

where $F_n$ denotes the $n$-th Fibonacci number, defined by $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_n$, $n\ge0$."

The chapter adds: "The bound (2) is best possible, as shown by the next result", Theorem 2. On p. 211 the theorem is restated as (34) with the same constant. The numerical value $\alpha=0.39441967\ldots$ is the chapter's; $1/\alpha=1+\sum_{k\ge1}F_{2k}^{-1}=2.535\ldots$ is the site's $c$ and the 1980 monograph's $\alpha_0$.

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 definition of $C$ and Theorem 1 on printed p. 182 (PDF p. 2 of the 42-page file, an image-only scan), the proof on printed p. 211 (PDF p. 31), read on the rendered page images. The edition is identified in the source digest. The same statement is Theorem 1 of the authors' 1981 announcement, theorem_1 of that card, which indexes the sequence from $x_0$.

Read depth. Claims checked: the definition of $C(\bar x)$, the theorem and the sentence after it were read clause by clause on the page image; the proof on p. 211 was read for its structure and Theorem 3, on which it rests, was not checked.

Proof pointer

P. 211: "As an immediate corollary of Theorem 3 we have: Theorem 1." Suppose (34) fails for some $\bar x$: then for every $n$ there is $\epsilon>0$ with $n|x_{m+n}-x_m|\ge(1+\epsilon)\alpha$ for all large $m$. By Theorem 3 (p. 183: the exact value of $u_m=\min_{\pi\in S_m}\max_I\sum_k|\pi(i_{k+1})-\pi(i_k)|^{-1}$, which tends to $1/\alpha$), for any $\delta>0$ and $N$ large every permutation $\pi\in S_N$ has an increasing subsequence $I=\{i_1<i_2<\cdots\}$ with $u(\pi)=\sum_k|\pi(i_{k+1})-\pi(i_k)|^{-1}>(1-\delta)/\alpha$. Let $\pi$ order $N$ consecutive terms $x_{M+1},\ldots,x_{M+N}$ ($M$ large). Then $1\ge\sum_k(x_{M+\pi(i_{k+1})}-x_{M+\pi(i_k)})\ge\alpha(1+\epsilon)\sum_k|\pi(i_{k+1})-\pi(i_k)|^{-1}>(1+\epsilon)(1-\delta)$, "which is a contradiction for $\delta$ sufficiently small." Theorem 3 itself occupies pp. 188--210 (the upper bound through the permutations $\rho_m$ induced by $\{k\tau\}$, pp. 188--203; the lower bound by induction on the statements $A(n)$, $B(n)$, $A'(n)$, $B'(n)$, pp. 203--210). Not reconstructed here.

Dependencies

Theorem 3 of the chapter (p. 183); the Fibonacci identities and approximation facts of the preliminaries (pp. 184--186); Lemmas 1--3 are cited only for Theorem 2 (pp. 182, 212--219). Self-contained otherwise.

Bears on

• Problem 480: the problem asks whether $\inf_n\liminf_{m\to\infty}n|x_{m+n}-x_m|\le5^{-1/2}\approx0.447$ for every sequence $x_1,x_2,\ldots\in[0,1]$; this is $C(\bar x)\le5^{-1/2}$, and Theorem 1 gives $C(\bar x)\le\alpha=0.3944\ldots<5^{-1/2}$, so the answer is yes with a smaller constant. The chapter's indexing from $x_1$ matches the problem's.