D-finiteness, rationality, and height II: Lower bounds over a set of positive density

We consider D-finite power series f(z) = sigma(n >= 0) a(n)z(n )with coefficients in a number field K. We show that there is a dichotomy governing the behaviour of h(a(n)) as a function of n, where h is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either f(z) is rational or h(a(n)) > [K : Q](-1) middot log(n) + O(1) for n in a set of positive upper density and this is best possible when K = Q.