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

TL;DR

This paper investigates the growth of the Weil height of coefficients of D-finite power series over number fields, establishing a dichotomy: such series are either rational or have coefficients with height growing at least logarithmically on a set of positive density.

Contribution

It proves a dichotomy for D-finite power series over number fields, showing either rationality or a lower bound on coefficient height growth on a positive density set.

Findings

Series are either rational or have height growth at least logarithmic.

Lower bounds on height growth are optimal for rational series over Q.

Results apply to coefficients in number fields with explicit density conditions.

Abstract

We consider D-finite power series $f(z)=\sum a_n z^n$ 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 $h(a_n)>[K:\mathbb{Q}]^{-1}\cdot \log(n)+O(1)$ for $n$ in a set of positive upper density and this is best possible when $K=\mathbb{Q}$.