  We determine a lower gap property for the growth of an unbounded
\(\mathbb{Z}\)-valued \(k\)-regular sequence. In particular, if
\(f:\mathbb{N}\to\mathbb{Z}\) is an unbounded \(k\)-regular sequence, we show
that there is a constant \(c>0\) such that \(|f(n)|>c\log n\) infinitely often.
We end our paper by answering a question of Borwein, Choi, and Coons on the
sums of completely multiplicative automatic functions.
