Year: 2025
Type: conference-paper
Abstract: We investigate the asymptotic growth of function CS(n), which maps n to the maximum number of distinct squares in a circular word of length n (that is, the maximum number of distinct squares of length at most n in a word ww of length 2n). We improve upon the lower bound of 1.25n established by Amit and Gawrychowski [SPIRE 2017] and the straightforward upper bound of 2n, which follows from the recent result of Brlek and Li [Comb. Theory, 2025] stating that there are fewer than n squares in standard (i.e., non-circular) words of length n. (Previously, Amit and Gawrychowski gave an upper bound of 3 2 15 n using a weaker upper bound on squares in standard words.) Specifically, we show that CS(n) ≤ ⌈1.8n⌉ and that, for infinitely many n, CS(n) ≥ 1.5n−O(√n). For the lower bound, we exploit the combinatorial structure of Fibonacci words to construct a family of square-rich circular words. For the upper bound, we exploit density properties of the starting positions of long squares, adapting an approach of Amit and Gawrychowski.
Institutions: King's College London, University of Warsaw
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7135164035
FWCI:
Cites:
Cited by:
Related to: