Abundance of arithmetic progressions in 𝒞ℛ-sets

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