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Assouad dimensions of complementary sets

Part of: Classical measure theory

Published online by Cambridge University Press:  12 April 2018

Ignacio García ,
Kathryn Hare  and
Franklin Mendivil
Ignacio García
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada ([email protected])
Kathryn Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada ([email protected])
Franklin Mendivil
Affiliation:
Department of Mathematics and Statistics, Acadia University, Wolfville, NS B4P 2R6, Canada ([email protected])
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Abstract

Given a positive, decreasing sequence a, whose sum is L, we consider all the closed subsets of [0, L] such that the lengths of their complementary open intervals are in one-to-one correspondence with the sequence a. The aim of this paper is to investigate the possible values that Assouad-type dimensions can attain for this class of sets. In many cases, the set of attainable values is a closed interval whose endpoints we determine.

Keywords

Assouad dimensioncomplementary setsCantor set

MSC classification

Secondary: 28A80: Fractals 28A78: Hausdorff and packing measures
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 3 , June 2018 , pp. 517 - 540
Copyright
Copyright © Royal Society of Edinburgh 2018 

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