论文标题
多相关序列和大回报的属性在某些真实性假设下
Properties of multicorrelation sequences and large returns under some ergodicity assumptions
论文作者
论文摘要
我们证明，给定一个度量保存系统$（x，x，\ nathcal {b}，μ，t_1，\ dots，t_d）$，并进行通勤，ergodic变换$ t_i $，因此$ t_it_jj^{ - 1} $对于所有$ i \ i \ i \ i \ neq j $，the neq j $，the neq j $，the neq j $，the neq j $ n cd y cd y cd n cd = t_1^nf_1 \ cdot \ dotso \ cdot t_d^n f_d \dμ$可以分解为$ a（n）= a _ {\ textrm {s st}}}}（n）+a _ {\ a _ {\ textrm {er}}（er}}}（n）$ a $ a nimim a n Limits a _ _ _ _ pem a i is a _ _ { $ d $ - 步骤nilSequences和$ a _ {\ textrm {er}} $是一个nullSequence（即，$ \ lim_ {n-m \ to \ infty} \ frac {1} {1} {n-m} {n-m} \ sum__________在$ t_1，\ dots，t_d $上的其他一些急性条件下，我们还为表单$ a（n）= \ int_x f_0 \ cdot \ cdot \ cdot \ prod_ = 1 = 1}^dt_i^^dt_i^^p_i^p_1} p_ {p_1} {i，1}}} cdot cdot的多项式多相关序列建立了类似的分解。 \ prod_ {i = 1}^dt_i^{p_ {i，k}（n）} f_k \dμ$，其中每个$ p_ {i，k}：\ mathbb {z} \ rightArrow \ rightArrow \ mathbb {z} $都是polynomial映射。我们还以$ d = 2 $的形式显示，如果$ t_1，t_2，t_1t_2^{ - 1} $是可逆且可恶的，我们有很大的三重交叉点：对于所有$ \ varepsilon> 0 $，以及所有$ a \ in \ mathcal {b} $ t_1^{ - n} a \ cap t_2^{ - n} a）>μ（a）^3- \ varepsilon \} $是syndetic。此外，我们表明，如果$ t_1，t_2，t_1t_2^{ - 1} $完全是ergodic，我们用$ p_n $ the $ n $ -th prime表示$ n $ \ \ \ {n \ in \ mathBb {n} n} t_2^{ - （p_n-1）} a）>μ（a）^3- \ varepsilon \} $具有较低的较低密度。
We prove that given a measure preserving system $(X,\mathcal{B},μ,T_1,\dots,T_d)$ with commuting, ergodic transformations $T_i$ such that $T_iT_j^{-1}$ are ergodic for all $i \neq j$, the multicorrelation sequence $a(n)=\int_X f_0 \cdot T_1^nf_1 \cdot \dotso \cdot T_d^n f_d \ dμ$ can be decomposed as $a(n)=a_{\textrm{st}}(n)+a_{\textrm{er}}(n)$, where $a_{\textrm{st}}$ is a uniform limit of $d$-step nilsequences and $a_{\textrm{er}}$ is a nullsequence (that is, $\lim_{N-M \to \infty} \frac{1}{N-M} \sum_{n=M}^{N-1} |a_{\textrm{er}}|^2=0$). Under some additional ergodicity conditions on $T_1,\dots,T_d$ we also establish a similar decomposition for polynomial multicorrelation sequences of the form $a(n)=\int_X f_0 \cdot \prod_{i=1}^dT_i^{p_{i,1}(n)}f_1\cdot\dotso \cdot \prod_{i=1}^dT_i^{p_{i,k}(n)}f_k \ dμ$, where each $p_{i,k}: \mathbb{Z} \rightarrow \mathbb{Z}$ is a polynomial map. We also show, for $d=2$, that if $T_1, T_2, T_1T_2^{-1}$ are invertible and ergodic, we have large triple intersections: for all $\varepsilon>0$ and all $A \in \mathcal{B}$, the set $\{n \in \mathbb{Z} : μ(A \cap T_1^{-n}A \cap T_2^{-n}A)>μ(A)^3-\varepsilon\}$ is syndetic. Moreover, we show that if $T_1, T_2, T_1T_2^{-1}$ are totally ergodic, and we denote by $p_n$ the $n$-th prime, the set $\{n \in \mathbb{N} : μ(A \cap T_1^{-(p_n-1)}A \cap T_2^{-(p_n-1)}A)>μ(A)^3-\varepsilon\}$ has positive lower density.
