Density of distinct squares in non-primitive words

Abstract

A word is said to be primitive if it cannot be expressed as a nontrivial power of another word. A repetition or a non-primitive word, uk, is a concatenation of two or more copies of a word u. A square is the smallest repetition of the form u2. We study the properties of distinct squares that are introduced when u2 is further extended with u′s. These properties are used to compute the density of distinct squares in a repetition. We show that the density of distinct squares in a repetition uk converges to 12 as k increases.