论文标题
用二进制单词的不同力量数量的上限
An upper bound of the number of distinct powers in binary words
论文作者
论文摘要
电源是$ \ underbrace {uu ... u} _ {k \; \ text {times}} $，其中$ u $是一个单词，$ k $是一个正整数，正方形是$ uu $的单词。 Fraenkel和Simpson在1998年猜想，单词中的不同正方形数量受单词长度的界定。这个猜想最近由Brlek和Li证明。此外，还有一个更强的上限，对于Jonoska，Manea和Seki猜想的二进制单词也更强大，指出，如果我们让$ n $长度$ n $的单词比字母$ \ left \ left \ {a，b \ right \} $，如果我们让$ k $是B的最低限度，然后是B的数量，然后是B的数量和$ K的数量，然后是GEQ 2 2 $ \ geq 2 2 $， $ \ frac {2k-1} {2k+2} n $。在本文中，我们通过对二进制单词中不同权力的数量提出更强有力的陈述来证明这种猜想。
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by the length of the word. This conjecture was proven recently by Brlek and Li. Besides, there exists a stronger upper bound for binary words conjectured by Jonoska, Manea and Seki stating that for a word of length $n$ over the alphabet $\left\{a, b\right\}$, if we let $k$ be the least of the number of a's and the number of b's and $k \geq 2$, then the number of distinct squares is upper bounded by $\frac{2k-1}{2k+2}n$. In this article, we prove this conjecture by giving a stronger statement on the number of distinct powers in a binary word.
