Topological correspondence of multiple ergodic averages of nilpotent group actions

Let (X,Γ) be a topological system, where Γ is a nilpotent group generated by T1,...,Td such that for each T ∈ Γ, T ≠ eΓ, (X,T) is weakly mixing and minimal. For d,k ∈ ℕ, let pi,j(n),1 ≤ i ≤ k,1 ≤ j ≤ d be polynomials with rational coefficients taking integer values on the integers and pi,j(0) = 0. We show that if the expressions \(g_i(n)=T_1^{{p}_{i,1}(n)}\cdots{T_d^{{p}_{i,d}(n)}}\) depend nontrivially on n for i = 1,2,...,k, and for all i ≠ j ∈ {1,2,...,k} the expressions gi(n)gj(n)-1 depend nontrivially on n, then there is a residual set X0 of X such that for all x ∈ X0 $$\{(g_1(n)x, g_2(n)x, ...., g_k(n)x)\in{X^k}:n\in\mathbb{Z}\}$$ is dense in Xk.