Finiteness of integral values for the ratio of two linear recurrences

Abstract

Let {F(n)}n∈N,{G(n)}n∈N, be linear recurrent sequences. In this paper we are concerned with the well-known diophantine problem of the finiteness of the set ? of natural numbers n such that F(n)/G(n) is an integer. In this direction we have for instance a deep theorem of van der Poorten; solving a conjecture of Pisot, he established that if ? coincides with N, then {F(n)/G(n)}n∈N is itself a linear recurrence sequence. Here we shall prove that if ? is an infinite set, then there exists a nonzero polynomial P such that P(n)F(n)/G(n) coincides with a linear recurrence for all n in a suitable arithmetic progression. Examples like F(n)=2n-2, G(n)=n+2n+(-2)n, show that our conclusion is in a sense best-possible. In the proofs we introduce a new method to cope with a notorious crucial difficulty related to the existence of a so-called dominant root. In an appendix we shall also prove a zero-density result for ? in the cases when the polynomial P cannot be taken a constant.