Integers representable as differences of linear recurrence sequences

TL;DR

This paper investigates the set of integers that can be expressed as differences of two linear recurrence sequences, providing an asymptotic count and showing that such integers have zero density.

Contribution

It establishes an asymptotic formula for the count of integers representable as differences of two linear recurrence sequences, revealing their zero density.

Findings

Asymptotic formula for the number of representable integers

Density of such integers is zero

Quantitative understanding of differences of recurrence sequences

Abstract

Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ 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 $0$.