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

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An infinitely generated self-similar set with positive Lebesgue measure and empty interior. / Baker, Simon; Sidorov, Nikita.

In: Proceedings of the American Mathematical Society, Vol. 147, 20.03.2019, p. 4891-4899.

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@article{ee6cbc5508e54469977ea247d32137fc,
title = "An infinitely generated self-similar set with positive Lebesgue measure and empty interior",
abstract = "In [Problems on self-similar sets and self-affine sets: An update, Birkh{\"a}user, 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 Cs{\"o}rnyei 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.",
author = "Simon Baker and Nikita Sidorov",
note = "First published in Proceedings of the American Mathematical Society in vol.147, 2019, published by the American Mathematical Society",
year = "2019",
month = mar,
day = "20",
doi = "10.1090/proc/14621",
language = "English",
volume = "147",
pages = "4891--4899",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
publisher = "American Mathematical Society",

}

RIS

TY - JOUR

T1 - An infinitely generated self-similar set with positive Lebesgue measure and empty interior

AU - Baker, Simon

AU - Sidorov, Nikita

N1 - First published in Proceedings of the American Mathematical Society in vol.147, 2019, published by the American Mathematical Society

PY - 2019/3/20

Y1 - 2019/3/20

N2 - 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.

AB - 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.

U2 - 10.1090/proc/14621

DO - 10.1090/proc/14621

M3 - Article

VL - 147

SP - 4891

EP - 4899

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

ER -