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

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 $I{P}^{\star }$-set. It is a well known and established result by Bergelson and Hindman that if $A$ is an $I{P}^{\star }$-set, then for any sequence $⟨x_n{⟩}_{n=1}^{\infty }$, there exists a sum subsystem $⟨y_n{⟩}_{n=1}^{\infty }$ such that $FS\left(⟨y_n{⟩}_{n=1}^{\infty }\right)\cup FP\left(⟨y_n{⟩}_{n=1}^{\infty }\right)\subset A$. 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 $I{P}^{\star }$-sets which are not equivalent to those of $I{P}^{\star }$-sets, but are rather significantly stronger. In this article, we will give the answer to Goswami's question that was unknown until now.

13 pages