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 $(X,\mathcal{B},\mu ),$ $q_1,\dots ,q_m:\mathbb{R}\to {\mathbb{R}}^{\ell }$ vector polynomials, and $f_0,\dots ,f_m\in L^{\infty }(X)$. For any $\epsilon >0$ and multicorrelation sequences of the form $\alpha (n)={\int }_X{f}_0\cdot T^{\lfloor q_1(n)\rfloor }{f}_1\cdots T^{\lfloor q_m(n)\rfloor }{f}_{m}d\mu$ we show that there exists a nilsequence $\psi$ for which $\underset{N-M\to \infty }{\lim }\frac{1}{N-M}\sum _{n=M}^{N-1}|\alpha (n)-\psi (n)|\le \epsilon$ and $\underset{N\to \infty }{\lim }\frac{1}{\pi (N)}\sum _{p\in \mathbb{P}\cap [1,N]}|\alpha (p)-\psi (p)|\le \epsilon .$ This result simultaneously generalizes previous results of Frantzikinakis [2] and the authors [11,13].

7 pages

References in corpus (6)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Integer part polynomial correlation sequences
• Weighted multiple ergodic averages and correlation sequences
• Nilsequences and Multiple Correlations along Subsequences
• Closest integer polynomial multiple recurrence along shifted primes
• A decomposition of multicorrelation sequences for commuting transformations along primes