Integers representable as differences of linear recurrence sequences

Abstract: Let ${U_n}{n \geqslant 0}$ and ${G_m}{m \geqslant 0}$ 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 $0$.