关于回归集与Van der Corput集的6个问题的解答
Solutions to 6 problems in recurrence and Van der Corput sets
- Beijing Institute of Mathematical Sciences and Applications（北京数学与应用研究所）
- The Ohio State University（俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决了关于优良回归集与van der Corput集的六个开放问题，证明了若干划分正则性及vdC集与算子回归集的关联，并由AI辅助生成初始证明后经人工验证与整理。
AI中文摘要：
我们给出了关于优良回归集与van der Corput（vdC）集的六个问题的解答。我们回答了Moreira提出的一个问题（Fish和Skinner也重复提及），通过证明每个遍历优良回归集都是优良回归集。我们回答了Bergelson以及Bergelson和Lesigne的问题，证明了每个优良回归集都是优良vdC集，优良vdC集族是划分正则的，并且优良回归集族也是划分正则的。我们提到Chapman独立证明了优良回归集是划分正则的。我们回答了Kelly和Lê的一个问题（其特殊情况先前由Peres提出），并证明对于每个非平凡紧致可度量群$K$，每个$K$-vdC集也是vdC集。最后，我们回答了Farhangi、Rodríguez和Tucker-Drob的一个问题，并证明对于任何可数无限离散群$G$，$G$中的vdC集是$G$中的算子回归集。这些问题的初始解答由人工智能系统产生。作者检查了其正确性，将其简化并重新表述为清晰、可读的证明，并将所得贡献置于现有文献背景中。
英文摘要：
We present solutions to six problems concerning sets of nice recurrence and van der Corput (vdC) sets. We answer a question of Moreira, repeated by Fish and Skinner, by showing that every set of ergodic nice recurrence is a set of nice recurrence. We answer questions of Bergelson as well as Bergelson and Lesigne by showing that every set of nice recurrence is a nice vdC-set, that the family of nice vdC-sets is partition regular, and that the family of sets of nice recurrence is partition regular. We mention that Chapman independently proved that sets of nice recurrence are partition regular. We answer a question of Kelly and Lê, a special case of which was previously asked by Peres, and show that for every nontrivial compact metrizable group $K$, every $K$-vdC set is also a vdC set. Lastly, we answer a question of Farhangi, Rodríguez, and Tucker-Drob and show that for any countably infinite discrete group $G$, a vdC set in $G$ is a set of operatorial recurrence in $G$. The initial solutions to these problems were produced by an AI system. The authors checked their correctness, simplified and reformulated them as clear, human-readable proofs, and situated the resulting contributions within the existing literature.
