An answer to Goswami's question and new sources of $IP^{\star}$-sets containing combined zigzag structure

TL;DR

This paper addresses Goswami's open question by exploring the existence of sum subsystems with combined zigzag finite sums and products within $IP^{ ext{star}}$-sets, extending known results in combinatorial number theory.

Contribution

It provides a positive answer to Goswami's question for $IP^{ ext{star}}$-sets, establishing new sources of combined zigzag structure in these sets.

Findings

Confirmed the existence of sum subsystems with zigzag structure in $IP^{ ext{star}}$-sets.

Extended the known results from dynamical $IP^{ ext{star}}$-sets to general $IP^{ ext{star}}$-sets.

Answered an open question posed by Goswami regarding combined zigzag structures.

Abstract

$A$ set is called $IP$-set in a semigroup $\left(S,\cdot \right)$ if it contains finite products of a sequence. A set that intersects with all $IP$-sets is called $IP^\star$-set. It is a well known and established result by Bergelson and Hindman that if $A$ is an $IP^{\star}$-set, then for any sequence $\langle x_{n}\rangle_{n=1}^{\infty}$, there exists a sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ such that $FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\cup FP\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\subset A$. In \cite[Question 3]{G}, S. Goswami posed the question: if we replace the single sequence by $l$-sequences, then is it possible to obtain a sum subsystem such that all of its zigzag finite sums and products will be in $A$. Goswami has given affirmative answers only for dynamical $IP^{\star}$-sets which are not equivalent to those of $IP^{\star}$-sets, but are rather significantly stronger. In this article, we will give the answer to Goswami's question that was unknown until now.