稀疏集上的多变量平均
Multivariable averaging on sparse sets
摘要：
非标准遍历平均可以定义为概率空间上群的保测度作用，作为经典（非标准）遍历平均的自然扩张。推广了一维理论，得到了几种非标准稀疏群平均的L^1逐点遍历定理，特别是对群Z^d。也就是说，我们将稀疏块平均和稀疏随机平均的结果推广到它们在虚幂零群上的类似物，并将Christ关于稀疏确定性序列的结果推广到它在Z^d上的类似物。第二个和第三个结果在Z^d上有两个非平凡的变量：一个“自然”d维平均值和一个来自1维平均值的“乘积”平均值。
英文摘要：
Nonstandard ergodic averages can be defined for a measure-preserving action of a group on a probability space, as a natural extension of classical (nonstandard) ergodic averages. We extend the one-dimensional theory, obtaining L^1 pointwise ergodic theorems for several kinds of nonstandard sparse group averages, with a special focus on the group Z^d. Namely, we extend results for sparse block averages and sparse random averages to their analogues on virtually nilpotent groups, and extend Christ's result for sparse deterministic sequences to its analogue on Z^d. The second and third results have two nontrivial variants on Z^d: a "native" d-dimensional average and a "product" average from the 1-dimensional averages.
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