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arXiv · 1708.06462

Constructing Words with High Distinct Square Densities

F. Blanchet-Sadri·S. Osborne
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Abstract

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 $2n-\Theta(\log n)$ 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 integer m, 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.

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F. Blanchet-Sadri, S. Osborne. 2017-08-22. Constructing Words with High Distinct Square Densities. https://doi.org/10.4204/eptcs.252.10

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