Integers representable as differences of linear recurrence sequences

Abstract

Let $\{U_n{\}}_{n⩾0}$ and $\{G_m{\}}_{m⩾0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x,x]$ which can be represented as differences $U_n-G_m$, when $x$ goes to infinity. In particular, the density of such integers is $0$.