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Distribution of integral values for the ratio of two linear recurrences

Abstract

Let FF and GG be linear recurrences over a number field K\mathbb{K}, and let R\mathfrak{R} be a finitely generated subring of K\mathbb{K}. Furthermore, let N\mathcal{N} be the set of positive integers nn such that G(n)0G(n) \neq 0 and F(n)/G(n)RF(n) / G(n) \in \mathfrak{R}. Under mild hypothesis, Corvaja and Zannier proved that N\mathcal{N} has zero asymptotic density. We prove that #(N[1,x])x(loglogx/logx)h\#(\mathcal{N} \cap [1, x]) \ll x \cdot (\log\log x / \log x)^h for all x3x \geq 3, where hh is a positive integer that can be computed in terms of FF and GG. Assuming the Hardy-Littlewood kk-tuple conjecture, our result is optimal except for the term loglogx\log \log x

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