Distribution of integral values for the ratio of two linear recurrences

Let F and G be linear recurrences over a number field K, and let R be a finitely generated subring of K. Furthermore, let N be the set of positive integers n such that G(n)≠0 and F(n)/G(n)∈R. Under mild hypothesis, Corvaja and Zannier proved that N has zero asymptotic density. We prove that #(N∩[1,x])≪x⋅(log⁡log⁡x/log⁡x)h for all x≥3, where h is a positive integer that can be computed in terms of F and G. Assuming the Hardy-Littlewood k-tuple conjecture, our result is optimal except for the term log⁡log⁡x.