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 a period of $i$. 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 $ 美元 。 所有单词长度 $ 的套套装 。 所有单词长度 $0, 1, 2,\ ldots, n-1 美元 的套装 。 但是, $ 0, 1, 2,\ ldots, n-1 $ 的任何组装 不一定是有效的套装 。 在1981 年的一份原始文件中, Guibas 和 Odlyzko 提议将单词的套装编码成 $ long 的二进制字符串, 称为自动关系, 其中, 以 $ 美元 的位值表示 $2 美元 的套装 。 他们审议了 承认一个有效期限 的问题, 并研究了 美元 $ 1, 1, 2, 2, 2, 2, 美元 美元 美元 美元 的套装配值 。 他们推测, 将单词的套装成一个固定时间 。 如果在2001 美元 ( kappappa) (n) 的更下框框框框框框框中,, 问题会持续到 。