迭代和集的维数增长
Dimension growth for iterated sumsets
摘要：
研究了和集和迭代和集的维数，给出了集合$F \subseteq \mathbb{R}$满足$\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$甚至$\dim_\text{H} n F \to 1$的自然条件.我们的结果适用于，例如，所有一致完美集，其中包括Ahlfors-David正则集。我们的证明依赖于Hochman的熵逆定理和Assouad和低维起着至关重要的作用。我们给出了我们的结果的几个应用，包括半群的Erdens-Schummann型定理和距离集的盒维数的新的下界的集合具有小的维数。
英文摘要：
We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$. Our results apply to, for example, all uniformly perfect sets, which include Ahlfors-David regular sets. Our proofs rely on Hochman's inverse theorem for entropy and the Assouad and lower dimensions play a critical role. We give several applications of our results including an Erdős-Volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
