与素数有关的数列的多重递归性和收敛性
Multiple recurence and convergence for sequences related to the prime numbers
摘要：
对任何保测度系统$(X,\mathcal{X},μ,T)$和$μ(A)>0$，证明了存在无穷多个素数$p$使得$μ\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$（同样的情形也适用于用$p+1$代替$p-1$）.此外，我们还证明了素数上的遍历平均在$L^2(μ)$中极限的存在性。一个关键的成分是最近的结果绿色和陶的冯曼戈尔特功能。一个组合的结果是，具有正的上密度的整数的每个子集包含长度为3的等差数列，并且对于某个素数$p$，其公差为$p-1$（或$p+1$）形式。
英文摘要：
For any measure preserving system $(X,\mathcal{X},μ,T)$ and $A\in\mathcal{X}$ with $μ(A)>0$, we show that there exist infinitely many primes $p$ such that $μ\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ (the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the existence of the limit in $L^2(μ)$ of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ (or $p+1$) for some prime $p$.
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