The upper density of an automatic set is rational

Given a natural number k ≥ 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 ≥ 2 and every pair of rational numbers ( α , β ) with 0 < α < β < 1 or with ( α , β ) ∈ { ( 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.