Nonstandard proof of combined sum and product structure in IP$^{*}$ sets

TL;DR

This paper provides a nonstandard analysis proof that IP* sets contain all finite sums and products of sum subsystems of any sequence in natural numbers, extending understanding of their combinatorial structure.

Contribution

It introduces a nonstandard analysis approach to prove a known combinatorial property of IP* sets, offering a new perspective and proof technique.

Findings

Nonstandard analysis can be used to prove properties of IP* sets.

IP* sets contain all finite sums and products of sum subsystems.

The proof offers a new method for analyzing combinatorial structures in number theory.

Abstract

V. Bergelson and N. Hindman proved that $IP^{*}$ sets contain all possible finite sum and product of a sum subsystem of any sequence in $\mathbb{N}$. In this article, we will prove this result using Nonstandard analysis.