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On the Hausdorff measure of shrinking target sets on self-conformal sets
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Allen, Demi and Bárány, Balázs (2021) On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika, 67 (4). pp. 807-839. doi:10.1112/mtk.12106 ISSN 0025-5793.
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Official URL: http://dx.doi.org/10.1112/mtk.12106
Abstract
In this article, we study the Hausdorff measure of shrinking target sets on self-conformal sets. The Hausdorff dimension of the sets we are interested in here was established by Hill and Velani in 1995 [Invent. Math. 119(1) (1995), 175–198]. However, until recently, little more was known about the Hausdorff measure of these particular sets. In this paper we provide a complete characterisation of the Hausdorff measure of these sets, obtaining a dichotomy for the Hausdorff measure which is determined by the convergence or divergence of a sum depending on the radii of our “shrinking targets”. Our main result complements earlier work of Levesley, Salp, and Velani [Math. Ann. 338(1) (2007), 97–118], and recent work of Baker [Math. Proc. Cambridge Philos. Soc. 167(3) (2019), 567–597] .
Item Type: | Journal Article | ||||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Journal or Publication Title: | Mathematika | ||||||||||||
Publisher: | London Mathematical Society | ||||||||||||
ISSN: | 0025-5793 | ||||||||||||
Official Date: | October 2021 | ||||||||||||
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Volume: | 67 | ||||||||||||
Number: | 4 | ||||||||||||
Page Range: | pp. 807-839 | ||||||||||||
DOI: | 10.1112/mtk.12106 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
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