Integers representable as differences of linear recurrence sequences

TL;DR

This paper investigates the distribution of integers that can be expressed as differences of two linear recurrence sequences, showing that such integers are extremely rare with zero density as the range expands.

Contribution

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

Findings

Number of representable integers grows slowly with range

Density of such integers approaches zero as range increases

Asymptotic formula characterizes the distribution of these integers

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 $0$.