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Cubefree binary words avoiding long squares

Narad Rampersad·Jeffrey Shallit·Ming-wei Wang
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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".

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Narad Rampersad, Jeffrey Shallit, Ming-wei Wang. 2003-04-07. Cubefree binary words avoiding long squares. https://arxiv.org/abs/math/0302303

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