Dimensions of popcorn-like pyramid sets

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-Erdos 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 Holder distortion between different graphs.

Automated summary

This article discusses the dimension theory of a family of functions' graphs, including the well-known 'popcorn function' and its higher-dimensional analogues. The box and Assouad dimensions, as well as the intermediate dimensions, which interpolate between Hausdorff and box dimensions, are calculated. Various tools, such as the Chung-Erdos inequality, higher-dimensional Duffin-Schaeffer type estimates, and Euler's totient function bounds, are used in the proofs. Applications include obtaining bounds on the box dimension of fractional Brownian images of the graphs and the Holder distortion between different graphs.