Dimension growth for iterated sumsets

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 ${\dim }_{\text{H}}nF\to 1$. 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.

23 pages, 2 figures. Minor changes. To appear in Math. Z