## Polynomial versus exponential growth in repetition-free binary words.(English)Zbl 1065.68080

Summary: 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 $$\frac{7}{3}$$. More precisely, there are only polynomially many binary words of length $$n$$ that avoid $$\frac{7}{3}$$-powers, but there are exponentially many binary words of length $$n$$ that avoid $$\frac{7}{3}^{+}$$-powers. This answers an open question of Kobayashi from 1986.

### MSC:

 68R15 Combinatorics on words
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### References:

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