Self similar sets, entropy and additive combinatorics
Self similar sets, entropy and additive combinatorics
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自相似集、熵和加性组合
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发表时间：
2013
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通讯作者：
M. Hochman
中科院分区：
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作者：
M. Hochman
本文阐述了[Hoc12]的主要结果，即维度小于平凡上界的自相似集在柱面之间“几乎重叠”。我们利用关于覆盖数的初等论据给出了该定理的一个启发式推导。我们还简要介绍了加法组合学，重点介绍了在证明中起关键作用的逆定理。我们的基本方法避免了[Hoc12]中的许多技术细节，但也缺乏完整的证明；在最后一节中，我们讨论如何将启发式论证转变为严格的论证。
This article is an exposition of the main result of [Hoc12], that self-similar sets whose dimension is smaller than the trivial upper bound have “almost overlaps” between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in [Hoc12], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.