Convergence of the number of period sets in strings

Consider words of length n. The set of all periods of a word of length n is a subset of {0, 1, 2, . . ., n−1}. However, any subset of {0, 1, 2, . . ., n−1} is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko 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 strings of length n, denoted κn. They conjectured that ln(κn) asymptotically converges to a constant times ln2(n). Although improved lower bounds for ln(κn)/ln2(n) were proposed in 2001, the question of a tight upper bound has remained open since Guibas and Odlyzko’s paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this longstanding 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