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

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Authors

Colleges, School and Institutes

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.

Bibliographic note

First published in Proceedings of the American Mathematical Society in vol.147, 2019, published by the American Mathematical Society

Details

Original languageEnglish
Pages (from-to)4891-4899
JournalProceedings of the American Mathematical Society
Volume147
Publication statusPublished - 20 Mar 2019