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