论非周期循环词的反幂
On Anti-Powers in Aperiodic Recurrent Words
摘要：
Fici、Restivo、Silva和Zamboni将$\textit{$ k $-anti-power}$定义为成对不同且具有相同长度的$k$连续字的级联。他们要求最大的$k$，使得每个非周期循环词必须包含一个$k$ -反幂，他们证明了这个最大值必须是3、4或5。我们通过证明最大值为5来解决这个问题。我们还推测，如果$W$是一个相当不错的非周期形态字，那么有一些恒定的$C = C(W)$，使所有的$i,k\geq 1$，$W$包含$k$ -反功率与块的长度最多$Ck$开始在其$i^\text{th}$的位置。我们解决了这个猜想的二进制字是由一个统一的态射生成的，其特点是一个小的例外的话，这样一个常数不能找到。这推广了最近的结果的第二作者，Gaetz和Narayanan已被证明的Thue-Morse字，这也表明，这样的线性界是最好的人可以希望在一般情况下。
英文摘要：
Fici, Restivo, Silva, and Zamboni define a $\textit{$k$-anti-power}$ to be a concatenation of $k$ consecutive words that are pairwise distinct and have the same length. They ask for the maximum $k$ such that every aperiodic recurrent word must contain a $k$-anti-power, and they prove that this maximum must be 3, 4, or 5. We resolve this question by demonstrating that the maximum is 5. We also conjecture that if $W$ is a reasonably nice aperiodic morphic word, then there is some constant $C = C(W)$ such that for all $i,k\geq 1$, $W$ contains a $k$-anti-power with blocks of length at most $Ck$ beginning at its $i^\text{th}$ position. We settle this conjecture for binary words that are generated by a uniform morphism, characterizing the small exceptional set of words for which such a constant cannot be found. This generalizes recent results of the second author, Gaetz, and Narayanan that have been proven for the Thue-Morse word, which also show that such a linear bound is the best one can hope for in general.
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。
