Integers representable as differences of linear recurrence sequences

Abstract

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