Combined exponential patterns in multiplicative $IP^{\star}$ sets

Abstract

$IP$ sets play fundamental role in arithmetic Ramsey theory. A set is called an additive $IP$ set if it is of the form $FS\left(\langle x_{n}\rangle_{n\in \mathbb{N}}\right)=\left\{ \sum_{t\in H}x_{t}:H\right.$ is a nonempty finite subset of $\left.\mathbb{N}\right\}$, whereas it is called a multiplicative $IP$ set if it is of the form $FP\left(\langle x_{n}\rangle_{n\in \mathbb{N}}\right)=\left\{ \prod_{t\in H}x_{t}:H\right.$ is a nonempty finite subset of $\left. \mathbb{N}\right\}$ for some injective sequence $\langle x_{n}\rangle_{n\in \mathbb{N}}.$ An additive $IP^{\star}$ (resp. multiplicative $IP^{\star}$) set is a set which intersects every additive $IP$ set (resp. multiplicative $IP$ set). In \cite{key-1}, V. Bergelson and N. Hindman studied how rich additive $IP^{\star}$ sets are. They proved additive $IP^{\star}$ sets ($AIP^{\star}$ in short) contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto in\cite{key-3}, where he proved that multiplicative $IP^{\star}$ sets ($MIP^{\star}$ in short) contain exponential tower\footnote{will be defined later} and finite product of a single sequence. However exponential patterns can be defined in two different ways. In this article we will prove that $MIP^{\star}$ sets contain two different exponential patterns and finite product of a single sequence. This immediately improves the result of A. Sisto. We also construct a $MIP^\star$ set, not arising from the recurrence of measurable dynamical systems. Throughout our work we will use the machinery of the algebra of the Stone-\v{C}ech Compactification of $\mathbb{N}$.