因子乘积中的重复避免
Repetition avoidance in products of factors
摘要：
我们考虑了Mousavi和Shallit在DLT 2013上介绍的关于单词的组合数学的经典回避问题的一个变体。令$\texttt{pexp}_i(w)$是词$w$的$i$因子的乘积上的指数的上确界。重复阈值$\texttt{RT}_i(k)$则是所有字$w\inΣ^ω_k$上的$\texttt{pexp}_i(w)$的下确界。Mousavi和Shallit获得了$\texttt{RT}_i(2)=2i$和$\texttt{RT}_2(3)=\tfrac{13}4$。我们证明了，如果$i$是偶数，则$\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac14$是$i$，如果$i$是奇数，则$\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac16$是$i$，而$i\ge3$是$i\ge3$。
英文摘要：
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 $w\inΣ^ω_k$. Mousavi 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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