THE MINIMAL GROWTH OF A -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.