Iterated compositions of linear operations on sets of positive upper density

TL;DR

This paper introduces iterated compositions of linear operations on sets of integers with positive upper density, proving their stability under certain conditions, extending previous difference sequence results.

Contribution

It generalizes the concept of iterated difference sequences to linear operations and establishes stability results for sets of positive upper density.

Findings

Proves stability of iterated linear compositions on dense integer sets

Extends previous difference sequence results to broader linear operations

Provides a general framework for analyzing linear transformations on dense sets

Abstract

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density.