December 2019
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126 Reads
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18 Citations
Mathematische Zeitschrift
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 $\overline{\dim_{\text{B}}} n F \to 1$. Our results apply to, for example, all uniformly perfect sets, which include Ahlfors-David regular sets. Our proofs rely on the inverse theorems of Bourgain and Hochman 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 dimensions of distance sets for sets with small dimension.