Structure of multicorrelation sequences with integer part polynomial iterates along primes

Abstract:

Let $T$ be a measure-preserving $\mathbb {Z}^\ell$-action on the probability space $(X,{\mathcal B},\mu ),$ let $q_1,\dots ,q_m\colon \mathbb {R}\to \mathbb {R}^\ell$ be vector polynomials, and let $f_0,\dots ,f_m \in L^\infty (X)$. For any $\epsilon > 0$ and multicorrelation sequences of the form $\alpha (n) =\int _Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;d\mu$ we show that there exists a nil- sequence $\psi$ for which $\lim _{N - M \to \infty } \frac {1}{N-M} \sum _{n=M}^{N-1} |\alpha (n) - \psi (n)| \leq \epsilon$ and $\lim _{N \to \infty } \frac {1}{\pi (N)} \sum _{p \in \mathbb {P}\cap [1,N]} |\alpha (p) - \psi (p)| \leq \epsilon .$ This result simultaneously generalizes previous results of Frantzikinakis and the authors.