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→Rℓ be vector polynomials, and let f0,…,fm∈L∞⁡(X). For any ϵ>0 and multicorrelation sequences of the form α⁡(n)=∫Xf0⋅T⌊q1⁡(n)⌋⁢f1⋯T⌊qm⁡(n)⌋⁢fmd⁢μ we show that there exists a nil- sequence ψ for which limN−M→∞⁡1N−M⁢∑n=MN−1|α⁡(n)−ψ⁡(n)|≤ϵ and limN→∞⁡1π⁡(N)⁢∑p∈P∩[1,N]|α⁡(p)−ψ⁡(p)|≤ϵ. This result simultaneously generalizes previous results of Frantzikinakis and the authors.