A height gap theorem for coefficients of Mahler functions

We study the asymptotic growth of coefficients Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. show that there are five different behaviors, all which being reached. Thus, gaps in possible growths. In proving this height gap theorem, we find a $k$-Mahler function is $k$-regular if and only its have $O(\log n)$. Moreover, deduce that, over an arbitrary ground field characteristic 0, $k$-automatic belong to finite set. As by-product our results, also recover conjecture Becker was recently settled Bell, Chyzak, Coons, Dumas.