On supra-SIM sets of natural numbers

TL;DR

This paper introduces supra-SIM sets of natural numbers, establishing their key properties like partition regularity and closure under finite-embeddability, and explores their sumset behavior inspired by Banach density concepts.

Contribution

It defines the class of supra-SIM sets and proves their fundamental properties, extending the understanding of SIM sets in additive number theory.

Findings

Supra-SIM sets are partition regular.

Supra-SIM sets are closed under finite-embeddability.

Results on sumsets of supra-SIM sets related to Banach density.

Abstract

We introduce the class of supra-SIM sets of natural numbers. We prove that this class is partition regular and closed under finite-embeddability. We also prove some results on sumsets and SIM sets motivated by their positive Banach density analogues.