Distribution of integral values for the ratio of two linear recurrences

Abstract

Let $F$ and $G$ be linear recurrences over a number field $\mathbb{K}$, and let $\mathfrak{R}$ be a finitely generated subring of $\mathbb{K}$. Furthermore, let $\mathcal{N}$ be the set of positive integers $n$ such that $G(n)\ne 0$ and $F(n)/G(n)\in \mathfrak{R}$. Under mild hypothesis, Corvaja and Zannier proved that $\mathcal{N}$ has zero asymptotic density. We prove that $\#(\mathcal{N}\cap [1,x])\ll x\cdot (\log \log x/\log x{)}^h$ for all $x\ge 3$, 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 $\log \log x$.

References in corpus (1)

• Roots of Sparse Polynomials over a Finite Field

Cited by in corpus (3)

• On the greatest common divisor of $n$ and the $n$th Fibonacci number
• On numbers $n$ relatively prime to the $n$th term of a linear recurrence
• On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence