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Almost sure Assouad-like dimensions of complementary sets

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Abstract

Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a. These are the so-called complementary sets, or rearrangements of the Cantor set, constructed from a . In this paper we determine the almost sure value of the \(\varPhi \)-dimension of these sets given a natural model of randomness. The \(\varPhi \)-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of \(\varPhi ,\) with one size behaving almost surely like the Assouad dimensions of the associated Cantor set and the other, like the quasi-Assouad dimensions. These results are new even for the Assouad dimensions.

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Notes

  1. The Cantor set associated with \(a=\{a_{j}\}\) is the set formed by first removing from [0, 1] an open interval of length \(a_{1}\) (whose position is uniquely determined by the construction), leaving two closed subintervals \(A_0\) (to the left) and \(A_1\) (to the right). Next, remove open intervals of lengths \(a_{2}\) and \(a_{3}\) from \(A_0\) and \(A_1\) respectively, leaving two closed subintervals in each of \(A_0\) and \(A_1\). Repeat this process in the usual Cantor construction fashion.

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Authors and Affiliations

  1. Facultad de Ciencias Exactas y Naturales, Centro Marplatense de Investigaciones Matemáticas, Instituto de Investigaciones Físicas de Mar del Plata (CONICET), Universidad Nacional de Mar del Plata, Mar del Plata, Argentina

    Ignacio García

  2. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Kathryn Hare

  3. Department of Mathematics and Statistics, Acadia University, Wolfville, NS, B4P 2R6, Canada

    Franklin Mendivil

Authors
  1. Ignacio García
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  2. Kathryn Hare
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  3. Franklin Mendivil
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Correspondence to Ignacio García.

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The research of K. Hare is partially supported by NSERC Discovery grant 2016:03719. The research of F. Mendivil is partially supported by NSERC Discovery grant 2019:05237. I. García and K. Hare thank Acadia University for their hospitality when some of this research was done. I. García thanks the hospitality of University of Waterloo.

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García, I., Hare, K. & Mendivil, F. Almost sure Assouad-like dimensions of complementary sets. Math. Z. 298, 1201–1220 (2021). https://doi.org/10.1007/s00209-020-02643-0

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