Constructing Words with High Distinct Square Densities

Fraenkel and Simpson showed that the number of distinct squares in a word of length n is bounded from above by 2n, since at most two distinct squares have their rightmost, or last, occurrence begin at each position. Improvements by Ilie to 2n−Θ(logn) and by Deza et al. to ⌊11n/6⌋ rely on the study of combinatorics of FS-double-squares, when the maximum number of two last occurrences of squares begin. In this paper, we first study how to maximize runs of FS-double-squares in the prefix of a word. We show that for a given positive integerm, the minimum length of a word beginning with m FS-double-squares, whose lengths are equal, is 7m+ 3. We construct such a word and analyze its distinct-square-sequence as well as its distinct-square-density. We then generalize our construction. We also construct words with high distinct-square-densities that approach 5/6.