论文标题
自动组的上部密度是理性的
The upper density of an automatic set is rational
论文作者
论文摘要
鉴于自然数量$ k \ ge 2 $和$ k $的自然数量$ s $ s $，我们表明$ s $的较低密度和上部密度是可递归计算的合理数字，我们为计算这些数量提供了算法。此外，我们表明，对于每个自然数量，$ k \ ge 2 $以及每对合理数量$（α，β）$，$ 0 <α<β<1 $或$（α，β）\ in \ {（0,0,0），（1,1）\} $ s $ k $ - $ k $ - uptomainty us naturaly naty and d $ d d.这些正是作为自动集合的下部密度和上部密度可能出现的值。
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.
