  We study the asymptotic growth of coefficients of Mahler power series with
algebraic coefficients, as measured by their logarithmic Weil height. We show
that there are five different growth behaviors, all of which being reached.
Thus, there are \emph{gaps} in the possible growths. In proving this height gap
theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its
coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an
arbitrary ground field of characteristic zero, a $k$-Mahler function is
$k$-automatic if and only if its coefficients belong to a finite set. As a
by-product of our results, we also recover a conjecture of Becker which was
recently settled by Bell, Chyzak, Coons, and Dumas.
