多重相关序列与 nilsequence
Multiple correlation sequences and nilsequences
AI总结：
本文证明可交换变换多项式迭代生成的多重相关序列可分解为 nilsequence 与均匀小误差之和，并据此推出线性迭代多重遍历平均的收敛准则。
摘要：
我们研究由可交换变换的保测作用所定义的多重相关序列的结构。当这些变换的迭代项为整系数多项式时，我们证明任何此类相关序列都可以表示为一个 nilsequence 与一个在均匀密度意义下很小的误差项之和；此前这一结论仅对单个变换的保测作用成立。随后，我们利用这一分解结果，给出涉及线性增长迭代项的多重遍历平均的收敛准则，并证明一个相当令人惊讶的事实：对于此类序列，可交换变换作用的收敛结果会自动由单个变换作用这一特殊情形推出。我们对分解结果的证明不同于 V. Bergelson、B. Host、B. Kra 和 A. Leibman 以往的工作，因为它不依赖特征因子理论。该证明由一个简单的正交性论证构成，主要工具是 B. Host 和 B. Kra 针对一般有界序列提出的逆定理。
英文摘要：
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 involving iterates that grow linearly, and prove the rather surprising fact that for such sequences, convergence results for actions of commuting transformations follow automatically from the special case of actions of a single transformation. Our proof of the decomposition result differs from previous works of V. Bergelson, B. Host, B. Kra, and A. 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 B. Host and B. Kra for general bounded sequences.
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