Multiple correlation sequences and nilsequences
Multiple correlation sequences and nilsequences
复制标题
多重相关序列和零序列
DOI：
10.1007/s00222-015-0579-7
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发表时间：
2014
影响因子：
3.1
通讯作者：
N. Frantzikinakis
中科院分区：
文献类型：
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作者：
N. Frantzikinakis
我们研究由通勤变换的测度保持动作定义的多个相关序列的结构。当变换的迭代是整数多项式时，我们证明任何这样的相关序列都是零序列和均匀密度较小的误差项的和；以前，这仅针对单个转换的度量保留操作而为人所知。然后，我们使用这个分解结果给出多个遍历平均值的收敛标准，并推导出一些相当令人惊讶的结果，例如，我们从单个变换动作的特殊情况推断出通勤变换动作的收敛性。我们对分解结果的证明与 Bergelson、Host、Kra 和 Leibman 之前的工作不同，因为它不依赖于特征因子理论。它由一个简单的正交性论证组成，主要工具是一般有界序列的 Host 和 Kra 的逆定理。
We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we prove that any such correlation sequence is the sum of a nilsequence and an error term that is small in uniform density; this was previously known only for measure preserving actions of a single transformation. We then use this decomposition result to give convergence criteria for multiple ergodic averages and deduce some rather surprising results, for instance we infer convergence for actions of commuting transformations from the special case of actions of a single transformation. Our proof of the decomposition result differs from previous works of Bergelson, Host, Kra, and Leibman, as it does not rely on the theory of characteristic factors. It consists of a simple orthogonality argument and the main tool is an inverse theorem of Host and Kra for general bounded sequences.