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An infinitely generated self-similar set with positive Lebesgue measure and empty interior


Authors: Simon Baker and Nikita Sidorov
Journal: Proc. Amer. Math. Soc. 147 (2019), 4891-4899
MSC (2010): Primary 28A80, 37C45
DOI: https://doi.org/10.1090/proc/14621
Published electronically: May 17, 2019
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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
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

DOI: https://doi.org/10.1090/proc/14621
Keywords: Self-similar sets, Lebesgue measure, interior
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
Article copyright: © Copyright 2019 American Mathematical Society