极小动力系统返回时间集的乘法组合性质
Multiplicative combinatorial properties of return time sets in minimal dynamical systems
摘要：
研究了极小拓扑动力系统的动力学性质与其返回时间集的乘法组合性质之间的关系。特别地，我们证明了对于任意极小系统中的剩余点集，返回到任意非空开集的时间集包含任意长的等比数列。在完全极小性和远距离性的假设下，证明了返回时间集沿$\mathbb{N}$和沿$\mathbb{N}$的乘法子半群分别具有正的乘法上Banach密度.这项工作的主要动机是长期悬而未决的问题，是否syndetic子集的正整数包含任意长的等比数列，我们的主要结果是一些证据肯定回答这个问题。
英文摘要：
We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.
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