The minimal density of a letter in an infinite ternary square - free word is 0 . 2746

We study the minimal density of letters in infinite square-free words. First, we give some definitions of minimal density in infinite words and prove their equivalence. Further, we propose a method that allows to strongly reduce an exhaustive search for obtaining lower bounds for minimal density. Next, we develop a technique for constructing square-free morphisms with extremely small density for one letter that gives upper bounds on the minimal density. As an application of our technique we prove that the minimal density of any letter in infinite ternary square-free words is 0.2746 · · ·. A word is called square-free if it cannot be written in the form axxb for two words a, b and a nonempty word x. It is easy to see that the maximal length of a binary square-free word is 3. A. Thue proved [8] that there exist ternary square-free words. The number of ternary square-free words of length n is given by the sequence A006156 in The Encyclopedia of Integer Sequences [7]. Ekhad and Zeilberger [2] proved that the number of ternary squarefree words of length n is at least 2. Grimm [3] gave a better bound; he proved that this number is at least 65. Note that not every finite square-free word can be extended to an infinite square-free word. In this paper we prove that the minimal density of any letter in an infinite ternary square-free word is 0.2746 · · ·.