Some negative answers to Bergelson–Hindman’s question

Let p1,…,pd be integral polynomials vanishing at 0. It was asked by Bergelson and Hindman whether, whenever A is large, the set {(m,n)∈N2:m+p1(n),m+p2(n),…,m+pd(n)∈A} is large in the same sense. In this paper, we give negative answers to this question when “large” is “central∗”, “IP∗”, “IP∗n”, “IP∗<ω” and “Δ∗”.