On the Density of Regular Languages

Abstract

Let ∑ be an alphabet which has at least two symbols. The density of L ⊆ ∑* is defined as D(L) := lim _n |L ∩ ∑ ⁿ |/|∑ ⁿ | ∈ [0, 1], provided that the limit exists. In 2015, R. Sin’ya has discovered an interesting relation between regular languages and their densities: If L ⊆ ∑* is a regular language, then D(L) = 0 if and only if there exists s ∈ ∑* such that ∑*s∑* ∩ L = ø. In this paper, we give a simple proof of this theorem, obtaining it as a simple consequence of the pumping lemma for regular languages.