线性子词复杂度的单词中Subword的频率
Frequencies of subwords in words of linear subword complexity
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中文总结 AI 辅助
本文利用Balková-Pelantová的方法，推导了线性子词复杂度单词的子词频率上界，并构造例子说明该上界无法进一步优化。
中文摘要 AI 辅助
使用Balková-Pelantová的方法，我们证明：若w是有限字母表上的右无限单词，则对每个非负整数N，w的长度为N+1的子词的不同上频率（若存在，下同）最多有3(p_w(N+1)-p_w(N))+1个，其中p_w(n)是w的子词复杂度函数。特别地，当w具有线性有界子词复杂度时，该结论给出一个统一上界。我们提供例子表明：只要f(n)是趋于无穷的弱增函数，就存在单词w，其长度为n的子词数量为O(nf(n))，且当N→∞时，w的长度为N的子词的不同上频率数量的上极限为无穷。
英文摘要
Using a method of Balková--Pelantová, we show that if ${\bf w}$ is a right-infinite word over a finite alphabet, then for each nonnegative integer $N$ there are at most $3(p_{\bf w}(N+1)-p_{\bf w}(N))+1$ distinct upper (and likewise lower and ordinary when they exist) frequencies for length-$(N+1)$ subwords of ${\bf w}$, where $p_{\bf w}(n)$ is the subword complexity function of $n$. In particular, this gives a uniform upper bound when ${\bf w}$ has linearly bounded subword complexity. We provide examples showing that whenever $f(n)$ is a weakly increasing function tending to infinity, there is a word ${\bf w}$ such that the number of subwords of length $n$ is $O(nf(n))$ and for which the limit supremum of the number of distinct upper frequencies of length-$N$ subwords of ${\bf w}$ as $N\to\infty$ is infinite.
