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Repetition avoidance in products of factors

Pamela Fleischmann Pascal Ochem 1 Kamellia Reshadi
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : We consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by Mousavi and Shallit at DLT 2013. Let $\texttt{pexp}_i(w)$ be the supremum of the exponent over the products of $i$ factors of the word $w$. The repetition threshold $\texttt{RT}_i(k)$ is then the infimum of $\texttt{pexp}_i(w)$ over all words $w\in\Sigma^\omega_k$. Moussavi and Shallit obtained that $\texttt{RT}_i(2)=2i$ and $\texttt{RT}_2(3)=\tfrac{13}4$. We show that $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac14$ if $i$ is even and $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac16$ if $i$ is odd and $i\ge3$.
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Pamela Fleischmann, Pascal Ochem, Kamellia Reshadi. Repetition avoidance in products of factors. Theoretical Computer Science, Elsevier, 2019, 791, pp.123-126. ⟨10.1016/j.tcs.2019.04.013⟩. ⟨lirmm-02500480⟩



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