Dimensions of popcorn-like pyramid sets
arXiv:2212.06961 · doi:10.4171/JFG/135
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$\unicode{x2013}$Erdős inequality from probability theory, higher-dimensional Duffin$\unicode{x2013}$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.
17 pages, 2 figures. Minor clarifications, results unchanged. To appear in the Journal of Fractal Geometry
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