Cubefree binary words avoiding long squares
Combinatorics
2007-05-23 v2 Discrete Mathematics
Abstract
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".
Cite
@article{arxiv.math/0302303,
title = {Cubefree binary words avoiding long squares},
author = {Narad Rampersad and Jeffrey Shallit and Ming-wei Wang},
journal= {arXiv preprint arXiv:math/0302303},
year = {2007}
}
Comments
PLEASE NOTE: After this paper was prepared, we learned that all our results appeared (albeit with different proofs) in a paper of F. M. Dekking, On repetitions of blocks in binary sequences, J. Combin. Theory Ser. A 20 (1976), 292--299