There are Binary Circular Cube-Free Words of Length $n$ Contained Within the Thue-Morse Word for all Positive Integers $n$.

Abstract

We extend the work of Currie and Fitzpatrick [1] on circular words avoiding patterns by showing that, for any positive integer $n$, the Thue-Morse word contains a subword of length $n$ which is circular cube-free. This proves a conjecture of V. Linek.