Structure of multicorrelation sequences with integer part polynomial iterates along primes

Let $T$ be a measure preserving $\mathbb{Z}^\ell$-action on the probability space $(X,{\mathcal B},μ),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$ vector polynomials, and $f_0,\dots,f_m\in L^\infty(X)$. For any $ε> 0$ and multicorrelation sequences of the form $\displaystyleα(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;dμ$ we show that there exists a nilsequence $ψ$ for which $\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |α(n) - ψ(n)| \leq ε$ and $\displaystyle\lim_{N \to \infty} \frac{1}{π(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |α(p) - ψ(p)| \leq ε.$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].

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