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Polynomial versus Exponential Growth in Repetition-Free Binary Words

Juhani Karhumaki·Jeffrey Shallit
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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+)-powers. This answers an open question of Kobayashi from 1986.

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Juhani Karhumaki, Jeffrey Shallit. 2003-04-07. Polynomial versus Exponential Growth in Repetition-Free Binary Words. https://arxiv.org/abs/math/0304095

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