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) not equal 0 and F(n)/G(n) is an element of R. Under mild hypothesis, Corvaja and Zannier proved that N has zero asymptotic density. We prove that #(N boolean AND [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. (C) 2017 Elsevier Inc. All rights reserved.