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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An infinitely generated self-similar set with positive Lebesgue measure and empty interior
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by Simon Baker and Nikita Sidorov PDF
Proc. Amer. Math. Soc. 147 (2019), 4891-4899 Request permission

Abstract:

In [Problems on self-similar sets and self-affine sets: An update, Birkhäuser, Basel, 2000] Peres and Solomyak ask the question: Do there exist self-similar sets with positive Lebesgue measure and empty interior? This question was answered in the affirmative by Csörnyei et al. in 2006. The authors of that paper gave a parameterised family of iterated function systems for which almost all of the corresponding self-similar sets satisfied the required properties. They did not however provide an explicit example. Motivated by a desire to construct an explicit example, we provide an explicit construction of an infinitely generated self-similar set with positive Lebesgue measure and empty interior.
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Additional Information
  • Simon Baker
  • Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
  • MR Author ID: 1001612
  • ORCID: 0000-0002-0716-6236
  • Email: simonbaker412@gmail.com
  • Nikita Sidorov
  • Affiliation: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
  • Email: sidorov@manchester.ac.uk
  • Received by editor(s): June 2, 2018
  • Received by editor(s) in revised form: February 20, 2019
  • Published electronically: May 17, 2019
  • Communicated by: Jeremy Tyson
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4891-4899
  • MSC (2010): Primary 28A80, 37C45
  • DOI: https://doi.org/10.1090/proc/14621
  • MathSciNet review: 4011521