Polynomial versus Exponential Growth in Repetition-Free Binary Words

TL;DR

This paper establishes that the growth rate of binary words avoiding certain fractional powers transitions from polynomial to exponential at the threshold of 7/3, resolving a long-standing open question.

Contribution

It precisely determines the critical exponent 7/3 as the dividing line between polynomial and exponential growth in repetition-free binary words, answering Kobayashi's 1986 question.

Findings

Binary words avoiding 7/3-powers grow polynomially.

Binary words avoiding more than 7/3-powers grow exponentially.

The critical threshold for growth rate change is exactly 7/3.

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.