# Regularity of infinite concatenation

It is well-known that an infinite union of regular languages is not necessarily regular, since every language can be written as a union of singletons. What about infinite concatenations?

Let $$\{ L_z : z \in \mathbb{Z} \}$$ be a family of languages, cofinitely many of which contain $$\epsilon$$. We can define the infinite concatenation of the family as the collection of all words of the form $$w_{i_1} \ldots w_{i_n}$$, where $$i_1 < \cdots < i_n$$ and $$w_{i_j} \in L_{i_j}$$. (The definition makes sense if we replace $$\mathbb{Z}$$ with any linear order.)

Is the infinite concatenation of a family of regular languages necessarily regular?

The answer is negative. Let $$L_n = \{\epsilon, a^{3^n}\}$$ for $$n \geq 0$$, and let $$L_n = \{\epsilon\}$$ for $$n < 0$$. The infinite concatenation of the collection $$\{ L_n : n \in \mathbb{Z} \}$$ is $$L = \{ a^m : \text{m is the sum of distinct powers of 3} \}.$$ Since $$L$$ is unary, in order to show that $$L$$ is not regular, we need to show that the set $$S = \{ m : \text{m is the sum of distinct powers of 3} \}$$ is not eventually periodic. We can see this in many ways. For example, $$S$$ is infinite but has density $$0$$ in the limit.