Dimension growth for iterated sumsets
Abstract
We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set F⊆ℝ satisfies ^dim^BF+F>^dim^BF or even dimHnF→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.
Citation
Fraser , J , Howroyd , D C & Yu , H 2018 , ' Dimension growth for iterated sumsets ' , Mathematische Zeitschrift , vol. First Online . https://doi.org/10.1007/s00209-018-2224-9
Publication
Mathematische Zeitschrift
Status
Peer reviewed
ISSN
0025-5874Type
Journal article
Rights
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Description
Jonathan M. Fraser was financially supported by a Leverhulme Trust Research Fellowship (RF-2016-500) and an EPSRC Standard Grant (EP/R015104/1). Douglas C. Howroyd was financially supported by the EPSRC Doctoral Training Grant (EP/N509759/1). Han Yu was financially supported by the University of St Andrews.Collections
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