The upper density of an automatic set is rational

TL;DR

This paper proves that the upper and lower densities of k-automatic sets of natural numbers are rational and computable, and characterizes all possible pairs of such densities.

Contribution

It establishes the rationality and computability of densities for automatic sets and characterizes all feasible density pairs.

Findings

Densities of automatic sets are rational and computable.

Algorithm provided for calculating densities.

All rational pairs within (0,1) or at endpoints are realizable as densities.

Abstract

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 $(\alpha,\beta)$ with $0<\alpha<\beta<1$ or with $(\alpha,\beta)\in \{(0,0),(1,1)\}$ there is a $k$-automatic subset of the natural numbers whose lower density and upper density are $\alpha$ and $\beta$ respectively, and we show that these are precisely the values that can occur as the lower and upper densities of an automatic set.