Dimension growth for iterated sumsets
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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.
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
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DescriptionJonathan 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.
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