Polynomial versus exponential growth in repetition-free binary words

Abstract

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3 : More precisely, there are only polynomially many binary words of length n that avoid 7 3 -powers, but there are exponentially many binary words of length n that avoid 7 3 þ -powers. This answers an open question of Kobayashi from 1986. r 2004 Elsevier Inc. All rights reserved.

Deciding context equivalence of binary overlap-free words in linear time

“…Proof (1) is obvious, a proof of (2) can be found in [13], (3) was first shown by Thue [15], so here we prove (4) and (5).…”

“…The language L α shares some key properties with the language OF, so it is quite probable that the context equivalence problem for L α can be decided in polynomial time by some modification of Algorithm E. We mention two such properties. First, the number of words in L α , as well as in OF, grows only polynomially with the length [5]. Second, the context equivalence problem for OF has such a low time complexity mainly because of the extremely small set of binary overlapfree morphisms.…”

On patterns occurring in binary algebraic numbers

“…The second main ingredient in the proof of Theorem 1.1 is the following structure theorem of Karhumäki and Shallit [5]. We mention that in the case where 7/3-powers are replaced by overlaps, this result was proved by Restivo and Salemi [8].…”

“…Our proof of Theorem 1.1 relies on two recent results: on one side, a combinatorial transcendence criterion due to Adamczewski, Bugeaud and Luca [2], which is based on the p-adic version of the Schmidt subspace theorem, and, on the other side, a structure theorem for binary words avoiding 7/3-powers obtained by Karhumäki and Shallit [5]. In fact, we will see that, quite surprisingly, infinite words avoiding 7/3-powers turn out to be rather repetitive in a different setting.…”

Growth of repetition-free words—a review

“…The proof is based on the following structural property of overlap-free words which we state in the more general setting of [22]. Recall first that the Thue-Morse morphism is defined by…”