论文标题
同时在NILSYSTEM中近似和回程集的乘法厚度
Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets
论文作者
论文摘要
在拓扑动力系统$（x，t）$中，如果存在一个序列$ n_1 $，$ n_2 $，... $ t^{n_i} x $，$ t $，$ t^$，$ t^{2n_i} x $，... 1978年，Furstenberg和Weiss表明，每个系统都具有同时近似自身（多重复发点）的点，并推断了Van der Waerden在算术进程上的改进。在本文中，我们研究了由给定点同时近似的一组点的密度。我们表明，在最小的NILSYSTEM中，所有点同时在一定限制的$ t $的功率集中近似于$δ$命中的积分。我们将该定理与返回时间集的多重组合特性联系起来，表明在最小系统中，所有NIL-BOHR集和典型的返回时间集都在自然数的乘法下群的固定位置上倍增。这产生了不均匀的多重复发结果，从而概括了Furstenberg和Weiss的定理，并导致了Van der Waerden定理的新增强。这项工作依赖于奥斯兰德，阿金和格拉斯纳开发的延长关系（轨道关联关系的封闭）的连续性。莱布曼，格林和陶（Tao）开发的Nilmanifolds的理性点和多项式理论；以及Glasner，Huang，Shao，Weiss和Ye最近开发的拓扑特征因素的机制。
In the topological dynamical system $(X,T)$, a point $x$ simultaneously approximates a point $y$ if there exists a sequence $n_1$, $n_2$, ... of natural numbers for which $T^{n_i} x$, $T^{2n_i}x$, ..., $T^{k n_i} x$ all tend to $y$. In 1978, Furstenberg and Weiss showed that every system possesses a point which simultaneously approximates itself (a multiply recurrent point) and deduced refinements of van der Waerden's theorem on arithmetic progressions. In this paper, we study the denseness of the set of points that are simultaneously approximated by a given point. We show that in a minimal nilsystem, all points simultaneously approximate a $δ$-dense set of points under a necessarily restricted set of powers of $T$. We tie this theorem to the multiplicative combinatorial properties of return-time sets, showing that all nil-Bohr sets and typical return-time sets in a minimal system are multiplicatively thick in a coset of a multiplicative subsemigroup of the natural numbers. This yields an inhomogeneous multiple recurrence result that generalizes Furstenberg and Weiss' theorem and leads to new enhancements of van der Waerden's theorem. This work relies crucially on continuity in the prolongation relation (the closure of the orbit-closure relation) developed by Auslander, Akin, and Glasner; the theory of rational points and polynomials on nilmanifolds developed by Leibman, Green, and Tao; and the machinery of topological characteristic factors developed recently by Glasner, Huang, Shao, Weiss, and Ye.
