Lower-bounds on the growth of power-free languages over large alphabets

TL;DR

This paper investigates the growth rates of power-free languages over large alphabets, proving a conjecture about their asymptotic lower bounds for certain parameters, thus advancing understanding of combinatorial properties of these languages.

Contribution

The paper proves the asymptotic lower-bound part of Shur's conjecture for the growth rate of power-free languages when 1<β<2, confirming the conjecture in this range.

Findings

Confirmed the asymptotic lower-bound of Shur's conjecture for 1<β<2.

Established the conjecture's validity for the case 9/8≤β<2.

Enhanced understanding of the growth behavior of power-free languages over large alphabets.

Abstract

We study the growth rate of some power-free languages. For any integer $k$ and real $\beta>1$, we let $\alpha(k,\beta)$ be the growth rate of the number of $\beta$-free words of a given length over the alphabet $\{1,2,\ldots, k\}$. Shur studied the asymptotic behavior of $\alpha(k,\beta)$ for $\beta\ge2$ as $k$ goes to infinity. He suggested a conjecture regarding the asymptotic behavior of $\alpha(k,\beta)$ as $k$ goes to infinity when $1<\beta<2$. He showed that for $\frac{9}{8}\le\beta<2$ the asymptotic upper-bound holds of his conjecture holds. We show that the asymptotic lower-bound of his conjecture holds. This implies that the conjecture is true for $\frac{9}{8}\le\beta<2$.