A concept of largeness of combinatorially rich sets

Abstract

In \cite[Proposition 8.21 Page-169]{F} Using the methods of topological dynamics, H. Furstenberg introduced the notion of central set and proved the famous Central Sets Theorem. Later, in \cite{DHS}, D. De, H. Hindman and D. Struss established a strong Central Sets Theorem, where they introduced the notion of $J$-set. Like $J$-set, in \cite{BG} V. Bergelson and D. Glasscock introduced the notion of combinatorially rich set ( $CR$-set). Let $u,v\in\mathbb{N}$ and $A$ be a $u\times v$ matrix with rational entries. In \cite{HS23}, N. Hindman and D. Strauss established that whenever $B$ is a piecewise syndetic set (resp. $J$-set) in $\mathbb{Z}$, $\left\{ \vec{x}\in\mathbb{Z}^{v}:A\vec{x}\in B^{u}\right\} $ is a piecewise syndetic set ( resp. $J$-set) in $\mathbb{Z}^{v}$. In this article, we prove the same result for $CR$-sets using an equivalent definition of $CR$-set in \cite{HHST} by N. Hindman, H. Hosseini, D. Strauss and M. Tootkaboni.