The minimal growth of a $k$-regular sequence

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.