循环避免阿贝尔幂
Avoiding abelian powers cyclically
摘要：
研究了交换幂循环回避的一个新概念。一个有限字$w$避免阿贝尔$N$ -幂循环，如果对于每个阿贝尔$N$ -幂周期$m$出现在无限字$w^ω$，我们有$m \geq |w|$。令$\mathcal{A}(k)$为最小整数$N$，使得对于所有$n$，在避免阿贝尔$N$ -幂循环的$k$-字母表上存在长度为$n$的字。令$\mathcal{A}_\infty(k)$是最小整数$N$，使得在$k$字母表上存在任意长的单词，这些单词循环地避免阿贝尔$N$幂。我们证明了对于$k \geq 5$，有$5 \leq \mathcal{A}(2) \leq 8$，$3 \leq \mathcal{A}(3) \leq 4$，$2 \leq \mathcal{A}(4) \leq 3$和$\mathcal{A}(k) = 2$。此外，我们表明，$\mathcal{A}_\infty(2) = 4$，$\mathcal{A}_\infty(3) = 3$，和$\mathcal{A}_\infty(4) = 2$。
英文摘要：
We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^ω$, we have $m \geq |w|$. Let $\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that there exist arbitrarily long words over a $k$-letter alphabet that avoid abelian $N$-powers cyclically. We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$, $2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$. Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) = 3$, and $\mathcal{A}_\infty(4) = 2$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
