Constructing Words with High Distinct Square Densities

TL;DR

This paper investigates the construction of words with high densities of distinct squares, providing bounds, explicit constructions, and analyzing their combinatorial properties to approach a density of 5/6.

Contribution

It introduces new constructions of words with maximized runs of FS-double-squares and high distinct-square-densities, advancing understanding of square-rich words.

Findings

Minimum length for words with m equal-length FS-double-squares is 7m+3

Constructed words with high distinct-square-densities approaching 5/6

Analyzed the distinct-square-sequence and density in constructed words

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.