平衡序列的渐近重复门限
Asymptotic repetitive threshold of balanced sequences
摘要：
有限字母表上无穷序列$\mathbf u$的临界指数$E(\mathbf u)$表示$\mathbf u$中因子的最大重复。利用著名的德让定理，对每一个$d$元序列$\mathbf u$都有$E(\mathbf u) \geq 1+\frac1{d-1}$.我们定义渐近临界指数$E^*(\mathbf u)$作为长度为$n$的因子的最大重复的上限。我们证明了对于任何$d>1$都存在一个$d$元序列$\mathbf u$，它的$E^*(\mathbf u)$任意接近于$1$ .然后重点研究了$d$元平衡序列类。在此类中，值$E^*(\mathbf u)$由严格大于1的阈值从下向上限定。我们提供了一种方法，使我们能够找到一个$d$元平衡序列的最小渐近临界指数为$2\leq d\leq 10$。
英文摘要：
The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary sequence $\mathbf u$. We define the asymptotic critical exponent $E^*(\mathbf u)$ as the upper limit of the maximal repetition of factors of length $n$. We show that for any $d>1$ there exists a $d$-ary sequence $\mathbf u$ having $E^*(\mathbf u)$ arbitrarily close to $1$. Then we focus on the class of $d$-ary balanced sequences. In this class, the values $E^*(\mathbf u)$ are bounded from below by a threshold strictly bigger than 1. We provide a method which enables us to find a $d$-ary balanced sequence with the least asymptotic critical exponent for $2\leq d\leq 10$.
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