标题： 沿形如$x^2 + ny^2$的素数的逐点遍历定理 显示英文标题

标题： Pointwise ergodic theorem along primes of the form $x^2 + ny^2$

摘要： 本文解决了沿形如$x^2 + ny^2$的素数多项式值集上单个函数的遍历平均值的逐点收敛问题。 遵循Bourgain的具有影响力的论文\cite{bourgain1989pointwise}，我们采用Hardy-Littlewood圆方法，其中素理想集的主要弧和次要弧估计是本文的主要创新点。 我们还证明了我们的收敛结果不能扩展到$L^1$函数类。 显示英文摘要

摘要： This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form $x^2 + ny^2$. Following the influential paper of Bourgain \cite{bourgain1989pointwise}, we employ the Hardy-Littlewood circle method where major arc and minor arc estimates for the set of prime ideals constitute the main novelty of the paper. We also prove that our convergence results cannot be extended to class of $L^1$ functions.