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Statement

For a real sequence $\bar x=(x_0,x_1,\ldots)$ with $x_i\in[0,1]$ the announcement measures its clustering by

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

As printed on p. 4001:

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

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

in which $F_n$ 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 bound 1 is best possible, as shown by the next result" (Theorem 2: $C(\bar x^*)=\alpha$ for the Fibonacci-digit sequence $\bar x^*$).

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 whole paper is this one printed page (PDF p. 1 of the one-page scan), read on the rendered page image. The edition is identified in the source digest.

Read depth. Claims checked: the definition of $C$ and the theorem were read clause by clause on the page image. The page gives no proof ("The proofs of the preceding results are somewhat delicate and rather lengthy and will be given elsewhere").

Proof pointer

None on the page. The proof is Theorem 1 of the 1984 chapter (p. 211 there, from its Theorem 3), which restates the result with the sequence indexed from $x_1$.

Dependencies

None stated on the page.

Bears on

• Problem 480: the problem's inequality is $C(\bar x)\le5^{-1/2}\approx0.447$ for sequences indexed from $x_1$; Theorem 1 gives $C(\bar x)\le0.3944\ldots$, and dropping or adding a first term does not change $C$, since the lower limit in $m$ ignores finitely many terms.