The quotient problem for linear recurrence sequences

TL;DR

This paper investigates the finiteness of ratios of linear recurrence sequences under certain algebraic conditions, employing advanced Diophantine approximation techniques.

Contribution

It establishes new finiteness results for ratios of linear recurrence sequences with algebraic constraints, using Schmidt's subspace theorem and moving hyperplanes.

Findings

Existence of a polynomial P such that d_{m,n}P(n)U(m)/V(n) is a multi-recurrence for m ≠ n.

V(n)/P(n) is a linear recurrence for m ≠ n.

Both d_{m,n}P(n)U(m)/V(n) and V(n)/P(n) are linear recurrences when m = n.

Abstract

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study the finiteness problem for the set $(m, n)\in \mathbb{N}^2$ such that there exist non-zero positive integers $d_{m, n}$ satisfying $\log |d_{m, n}|=o(n)$, and $d_{m, n}U(m)/V(n)$ is an element from a finitely generated subring of $\C$. In particular, we prove that for $m\neq n $, there exists a polynomial $P$ such that $d_{m, n}P(n)U(m)/V(n)$ is a multi-recurrence and $V(n)/P(n)$ is a linear recurrence and for $m=n$ both $d_{m, n}P(n)U(m)/V(n)$ and $V(n)/P(n)$ are linear recurrences. To prove our results, we employ Schmidt's subspace theorem, and the concept of moving hyperplanes, moving polynomials, and moving points.