🤖 AI Summary
This work investigates the growth rate of the language $mathcal{L}(mathcal{A},mathcal{F})$ over alphabet $mathcal{A}$ avoiding a forbidden factor set $mathcal{F}$, aiming to establish an explicit, decidable, and constructive sufficient condition for **bounded supermultiplicativity**—i.e., growth asymptotically bounded between $alpha^n$ and $Calpha^n$. The approach integrates combinatorics on words, symbolic dynamical systems, and asymptotic enumeration techniques. It yields the first algorithmically verifiable criterion for this property and extends the analysis to power-free and circular (necklace) words. Key contributions are: (1) effective computation of the growth rate $alpha$ and the bounding constant $C$; (2) tighter growth bounds for power-free words; and (3) new results on counting square-free circular words, partially confirming Shur’s conjecture.
📝 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 $alphage0$, such that for all $n$, the number of words of length $n$ in $mathcal{L}(mathcal{A},mathcal{F})$ is between $alpha^n$ and $Calpha^n$. In some settings, our condition provides a way to compute $C$, which implies that $alpha$, 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.
