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 arbitr...

We propose a notion of functor points for noncommutative spaces, valued in categories bimodules, and endowed with an action functional determined by hermitian structures height functions, modeled on interpretation the classical as physical sigma model. discuss different choices such based notions volumes traces, including one Hattori-Stallings rank. show that function determines dynamical time ...