An explicit condition for boundedly supermultiplicative subshifts
Vuong Bui, Matthieu Rosenfeld
TL;DR
This work identifies explicit, computable conditions on an alphabet $\mathcal{A}$ and a forbidden-factor set $\mathcal{F}$ that guarantee boundedly supermultiplicative growth for the language $\mathcal{L}(\mathcal{A},\mathcal{F})$ and its growth rate $\alpha(\mathcal{A},\mathcal{F})$. By deriving a precise lower bound $|\mathcal{L}_{n+m}| \ge C\,|\mathcal{L}_n|\,|\mathcal{L}_m|$ under a growth condition with a computable constant $C$, the authors connect submultiplicativity, analytic criteria via $\omega(x)$, and algorithmic computability of $\alpha(\mathcal{A},\mathcal{F})$. They adapt the approach to $p$-power-free languages, obtaining sharper bounds and, in the square-free case, explicit constants, including results on the growth of square-free circular words for alphabets of size at least $5$. Extending these ideas to circular words and generalized $\mathcal{F}$-free circular words, they show that circular versions often share the same exponential growth rate as their linear counterparts, and discuss the Restivo–Salemi property and connections to Shur’s conjecture. Overall, the paper provides concrete, computable tools to bound and approximate growth rates in complex avoidance languages and advances understanding of circular-avoidance phenomena.
Abstract
We study some properties of the growth rate of $\mathcal{L}(\mathcal{A},\mathcal{F})$, that is, the language of words over the alphabet $\mathcal{A}$ avoiding the set of forbidden factors $\mathcal{F}$. We first provide a sufficient condition on $\mathcal{F}$ and $\mathcal{A}$ for the growth of $\mathcal{L}(\mathcal{A},\mathcal{F})$ to be boundedly supermultiplicative. That is, there exist constants $C>0$ and $α\ge0$, such that for all $n$, the number of words of length $n$ in $\mathcal{L}(\mathcal{A},\mathcal{F})$ is between $α^n$ and $Cα^n$. In some settings, our condition provides a way to compute $C$, which implies that $α$, the growth rate of the language, is also computable whenever our condition holds. We also apply our technique to the specific setting of power-free words where the argument can be slightly refined to provide better bounds. Finally, we apply a similar idea to $\mathcal{F}$-free circular words and in particular we make progress toward a conjecture of Shur about the number of square-free circular words.