Abundance of arithmetic progressions in $\mathcal{CR}$-sets

Abstract

H.Furstenberg and E.Glasner proved that for an arbitrary $k\in\mathbb{N}$, any piecewise syndetic set of integers contains a $k$-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in $\mathbb{Z}.$ The above result was extended for arbitrary semigroups by V. Bergelson and N. Hindman, using the algebra of the Stone-\v{C}ech compactification of discrete semigroups. However, they provided an abundance for various types of large sets. In \cite{DHS}, the first author, Neil Hindman and Dona Strauss introduced two notions of large sets, namely, $J$-set and $C$-set. In \cite{BG}, V. Bergelson and D. Glasscock introduced another notion of largeness, which is analogous to the notion of $J$-set, namely $\mathcal{CR}$- set. All these sets contain arithmetic progressions of arbitrary length. In \cite{DG}, the second author and S. Goswami proved that for any $J$-set, $A\subseteq\mathbb{N}$, the collection $\{(a,b):\,\{a,a+b,a+2b,\ldots,a+lb\}\subset A\}$ is a $J$-set in $(\mathbb{N\times\mathbb{N}},+)$. In this article, we prove the same for $\mathcal{CR}$-sets.