Fife's Theorem for (7/3)-Powers

An overlap is a word of the form axaxa, where a is a single letter and x is a (possibly empty) word. In 1980, Earl Fife [8] proved a theorem characterizing the infinite binary overlap-free words as encodings of paths in a finite automaton. Berstel [4] later simplified the exposition, and both Carpi [6] and Cassaigne [7] gave an analogous analysis for the case of finite words. In a previous paper [13], the second author gave a new approach to Fife’s theorem, based on the factorization theorem of Restivo and Salemi [12] for overlap-free words. In this paper, we extend this analysis by applying it to the case of 3 -power-free words. Given a rational number p q > 1, we define a word w to be a p q -power if w can be written in the form xnx′ where n = ⌊p/q⌋, x′ is a (possibly empty) prefix of x, and |w|/|x| = p/q. The word x is called a period of w, and p/q is an exponent of w. If p/q is the largest exponent of w, we write exp(w) = p/q. We also say that w is |x|-periodic. For example, the word alfalfa is a 7 3 -power, and the corresponding period is alf. Sometimes, as is routine in the literature, we also refer to |x| as the period; the context should make it clear which is meant. A word, whether finite or infinite, is β -power-free if it contains no factor w that is an α-power for α ≥ β . A word is β+-power-free if it contains no factor w that is an α-power for α > β . Thus, the concepts of “overlap-free” and “2+-power-free” coincide.