Consider words of length $n$. The set of all periods of a word of length $n$ is a subset of $\{0,1,2,\ldots,n-1\}$. However, any subset of $\{0,1,2,\ldots,n-1\}$ is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko have proposed to encode the set of periods of a word into an $n$ long binary string, called an autocorrelation, where a one at position $i$ denotes the period $i$. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for length $n$, denoted $\kappa_n$. They conjectured that $\ln(\kappa_n)$ asymptotically converges to a constant times $\ln^2(n)$. If improved lower bounds for $\ln(\kappa_n)/\ln^2(n)$ were proposed in 2001, the question of a tight upper bound has remained opened since Guibas and Odlyzko's paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this long standing conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings.

翻译：考虑长度为 $n$ 的单词。长度为 $n$ 的单词的所有周期集合是 $\{0,1,2,\ldots,n-1\}$ 的子集。然而，$\{0,1,2,\ldots,n-1\}$ 的任意子集并不一定是有效的周期集合。在1981年的一篇开创性论文中，Guibas 和 Odlyzko 提出将单词的周期集合编码为一个长度为 $n$ 的二进制字符串，称为自相关，其中位置 $i$ 处的 1 表示周期 $i$。他们研究了识别有效周期集合的问题，并探讨了长度为 $n$ 的有效周期集合数量（记为 $\kappa_n$）。他们猜想 $\ln(\kappa_n)$ 渐近收敛于一个常数乘以 $\ln^2(n)$。尽管在2001年提出了 $\ln(\kappa_n)/\ln^2(n)$ 的改进下界，但自 Guibas 和 Odlyzko 的论文以来，紧上界问题一直悬而未决。本文我们给出了该比值的上界，从而证明了其收敛性并解决了这一长期存在的猜想。此外，我们将结果推广到互相关数量的类似界：互相关是自相关的一种推广，用于编码两个字符串之间的重叠关系。