Dimensions of popcorn-like pyramid sets
Amlan Banaji
University of St Andrews, UKHaipeng Chen
Shenzhen Technology University, China
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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 dimensions. 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.
Cite this article
Amlan Banaji, Haipeng Chen, Dimensions of popcorn-like pyramid sets. J. Fractal Geom. 10 (2023), no. 1/2, pp. 151–168
DOI 10.4171/JFG/135