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

Abstract

We consider D-finite power series f(z)=∑n≥0anzn with coefficients in a number field K. We show that there is a dichotomy governing the behaviour of h(an) 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(an)>[K:Q]−1⋅log⁡(n)+O(1) for n in a set of positive upper density and this is best possible when K=Q.