Structure of multicorrelation sequences with integer part polynomial iterates along primes

Let T be a measure-preserving Zℓ-action on the probability space (X, B, μ), let q1, . . . , qm: R → R_ be vector polynomials, and let f0, . . . ,fm ∈ L∞(X). For any ∞ > 0 and multicorrelation sequences of the form α(n) = ∞X f0 T∞q1(n)∞f1 T∞qm(n)∞fm dμ we show that there exists a nilsequence ψ for which limN-M→∞ 1 N-M -N-1 n=M |α(n) - ψ(n)| ≤ ϵ and limN→∞ 1 π(N) ϵp∈P∩[1,N] |α(p) - ψ(p)| ≤ ϵ. This result simultaneously generalizes previous results of Frantzikinakis and the authors.