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arXiv · 1802.03324

Dimension growth for iterated sumsets

Jonathan M. Fraser·Douglas C. Howroyd·Han Yu
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Abstract

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 Erd\H{o}s-Volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension.

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Jonathan M. Fraser, Douglas C. Howroyd, Han Yu. 2018-02-09. Dimension growth for iterated sumsets. https://arxiv.org/abs/1802.03324

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