Cubefree words with many squares

Cubefree binary words avoiding long squares

Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares xx with |x| ≥ 4, and the number 4 is best possible. However, the Entringer-Jackson-Schatz conjecture is true if “cubefree” is replaced with “overlap-free”.

There are k-uniform cubefree binary morphisms for all k >= 0

A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Σ * → Σ * is k-uniform if h(a) has length k for all a ∈ Σ. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k ≥ 0 there exists a k-uniform cubefree binary morphism.

On the Entropy and Letter Frequencies of Powerfree Words

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.

Polynomial versus exponential growth in repetition-free binary words

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 73 . 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 avo...

Cubefree Construction Algorithm for Cubefree Groups A GAP 4 Package