论文标题
平衡序列的渐近重复阈值
Asymptotic repetitive threshold of balanced sequences
论文作者
论文摘要
有限字母上无限序列$ \ mathbf u $的关键指数$ e（\ mathbf u）$表达了$ \ mathbf u $中一个因子的最大重复。由著名的DeJean定理，$ e（\ Mathbf U）\ geq 1+ \ frac1 {d-1} $每$ d $ -ary序列$ \ mathbf u $。我们将渐近关键指数$ e^*（\ mathbf u）$定义为长度$ n $的最大重复的上限。我们证明，对于任何$ d> 1 $，存在$ d $ - ary序列$ \ mathbf u $，其$ e^*（\ mathbf u）$任意接近$ 1 $。然后，我们专注于$ d $ - ary平衡序列的类。在此类中，值$ e^*（\ mathbf u）$从下面的阈值严格大于1的阈值。我们提供了一种方法，使我们能够找到一个$ d $ ar的平衡序列，最小渐近的关键指标，价格为$ 2 \ leq d \ leq d \ leq d \ leq 10 $。
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 $2\leq d\leq 10$.
