Title: Constructing Words with High Distinct Square Densities,
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.