The fixed length Levenshtein (FLL) distance between two words $\mathbf{x,y} \in \mathbb{Z}_m^n$ is the smallest integer $t$ such that $\mathbf{x}$ can be transformed to $\mathbf{y}$ by $t$ insertions and $t$ deletions. The size of a ball in FLL metric is a fundamental but challenging problem. Very recently, Bar-Lev, Etzion, and Yaakobi explicitly determined the minimum, maximum and average sizes of the FLL balls with radius one. In this paper, based on these results, we further prove that the size of the FLL balls with radius one is highly concentrated around its mean by Azuma's inequality.

翻译：两个单词$\mathbf{x,y} \in \mathbb{Z}_m^n$之间的固定长度莱文斯坦（FLL）距离是将$\mathbf{x}$通过$t$次插入和$t$次删除操作转换为$\mathbf{y}$所需的最小整数$t$。FLL度量中球的大小是一个基础但具有挑战性的问题。近期，Bar-Lev、Etzion和Yaakobi明确确定了半径为1的FLL球的最小、最大和平均大小。本文基于这些结果，利用Azuma不等式进一步证明了半径为1的FLL球的大小高度集中在其均值附近。