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 $\dim_\text{H} n F \to 1$. 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.