We characterize the squares occurring in infinite overlap-free binary words and construct various α power-free binary words containing infinitely many overlaps. 1

Abstract. We prove two cases of a strong version of Dejean’s conjecture involving “ extremal letter frequencies. The results are that there exist an 5+ infinite 4-free word over a 5 letter alphabet with letter frequency and an infinite-free word over a 6 letter alphabet with letter

Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sucient to construct an in nite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that this constant is 0:55080:::.

We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80. 1