Dimensions of popcorn-like pyramid sets
Abstract
This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung–Erdős inequality from probability theory, higher-dimensional Duffin–Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the Hölder distortion between different graphs.
Citation
Banaji , A & Chen , H 2023 , ' Dimensions of popcorn-like pyramid sets ' , Journal of Fractal Geometry , vol. 10 , no. 1 , pp. 151-168 . https://doi.org/10.4171/jfg/135
Publication
Journal of Fractal Geometry
Status
Peer reviewed
ISSN
2308-1309Type
Journal article
Rights
Copyright © 2023 European Mathematical Society. Published by EMS Press. This work is licensed under a CC BY 4.0 license.
Description
AB was financially supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). HC was supported by NSFC (No. 11871227) and Shenzhen Science and Technology Program (Grant No. RCBS20210706092219049).Collections
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