# Distribution of integral values for the ratio of two linear recurrences

## TL;DR

This paper investigates the distribution of integral values of the ratio of two linear recurrences over a number field, providing bounds on their density and assuming conjectures for optimality.

## Contribution

It establishes explicit bounds on the count of integers where the ratio of two linear recurrences lies in a finitely generated subring, extending prior zero-density results.

## Key findings

- Bound on the number of such integers: x imes (rac{\u2212 ext{log log x}}{ ext{log x}})^h
- Result holds under mild hypotheses and is nearly optimal under Hardy-Littlewood conjecture
- Provides a quantitative measure of the distribution of ratios of 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 $G(n) \neq 0$ 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 $x \geq 3$, 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 $\log \log x$.