A height gap theorem for coefficients of Mahler functions

Abstract

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 gaps in the possible growths. In proving this height gap theorem, we find that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Moreover, we deduce that, over an arbitrary ground field of characteristic 0, 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.