自动机组的上密度是合理的
The upper density of an automatic set is rational
摘要：
给定一个自然数$k\ge 2$和一个自然数的$k$ -自动集$S$，我们证明了$S$的下密度和上密度是递归可计算的有理数，并给出了计算这些量的算法.此外，我们证明了，对于每个自然数$k\ge 2$和每对有理数$(α,β)$与$0<α<β<1$或与$(α,β)\in \{(0,0),(1,1)\}$有一个$k$ -自动子集的自然数，其下密度和上密度分别是$α$和$β$，我们表明，这些正是值，可以出现的自动集的下密度和上密度。
英文摘要：
Given a natural number $k\ge 2$ and a $k$-automatic set $S$ of natural numbers, we show that the lower density and upper density of $S$ are recursively computable rational numbers and we provide an algorithm for computing these quantities. In addition, we show that for every natural number $k\ge 2$ and every pair of rational numbers $(α,β)$ with $0<α<β<1$ or with $(α,β)\in \{(0,0),(1,1)\}$ there is a $k$-automatic subset of the natural numbers whose lower density and upper density are $α$ and $β$ respectively, and we show that these are precisely the values that can occur as the lower and upper densities of an automatic set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
