As we have seen (Theorem 5.4.2), every ultimately periodic sequence is k-automatic for all integers k ≥ 2. In this chapter we prove a beautiful and deep theorem due to Cobham, which states that if a sequence s = (s(n))n≥0 is both k-automatic and l-automatic and k and l are multiplicatively independent, then S is ultimately periodic. (Recall that Theorem 2.5.7 discusses when two integers are multiplicatively independent.)

Syndetic and Right Dense Sets

In this section, we prove some useful preliminary results.

We say that a set X ⊆ Σ* is right dense if for any word u ∈ Σ* there exists a υ ∈ Σ* such that uυ ∈ X (that is, any word appears as a prefix of some word in X).

Lemma 11.1.1Let k, l ≥ 2 be multiplicatively independent integers, and let X be an infinite k-automatic set of integers. Then 0*(X)l = 0*{(n)l : n ∈ X} is right dense.

Proof. Since X is infinite and k-automatic, by the pumping lemma there exist strings t, u, υ with u nonempty such that tu*υ ⊆ (X)k. Let x ∈ {0, 1, …, l — 1}*. Our goal is to construct y such that xy ∊ 0*(X)l.