整数部分多项式沿素数迭代的多相关序列的结构
Structure of multicorrelation sequences with integer part polynomial iterates along primes
摘要：
令$T$是概率空间$(X,{\mathcal B},μ),$ $q_1,\dots,q_m:{\mathbb R}\to{\mathbb R}^\ell$向量多项式和$f_0,\dots,f_m\in L^\infty(X)$上的保持$\mathbb{Z}^\ell$ -作用的测度。对任意$ε> 0$和$\displaystyleα(n)=\int_Xf_0\cdot T^{ \lfloor q_1(n) \rfloor }f_1\cdots T^{ \lfloor q_m(n) \rfloor }f_m\;dμ$形式的多重相关序列，我们证明了存在一个零序列$ψ$，使$\displaystyle\lim_{N - M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |α(n) - ψ(n)| \leq ε$和$\displaystyle\lim_{N \to \infty} \frac{1}{π(N)} \sum_{p \in {\mathbb P}\cap[1,N]} |α(p) - ψ(p)| \leq ε.$满足.
英文摘要：
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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