The square density of words having a sequence of FS-double squares

Abstract

A square is a word of the form uu, where u is a non-empty word. The square density of a word is the ratio of the number of distinct squares in a word to its length. It is known that in a word w=a1a2…an, at most two different rightmost squares can start from a letter ai. A factor of w that begins with two rightmost squares in w is said to be an FS-double square. We identify the structure of words having k consecutive FS-double squares in which at least k locations start with no rightmost squares. We show that the square density of such words is less than 13381. We select the structure of FS-double squares that minimizes the number of locations starting with no rightmost squares and use it to obtain a new family of words. Every word of the family has consecutive FS-double squares. We then prove that the square density of these words increases with the size of the words and converges to one.