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$\unicode{x2013}$Erd\H{o}s 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 H\"older distortion between different graphs.