The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

A word on a finite alphabet A is said to be overlap-free if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlap-free binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlap-free words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlap-free words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...

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