Dimensions of popcorn-like pyramid sets

TL;DR

This paper investigates the fractal dimensions of graphs of popcorn-like functions and their higher-dimensional analogues, providing explicit calculations and bounds using advanced mathematical tools.

Contribution

It introduces the calculation of various fractal dimensions for these functions' graphs, including intermediate dimensions, and applies probabilistic and number-theoretic methods.

Findings

Calculated box and Assouad dimensions of the graphs

Derived bounds on fractional Brownian images of the graphs

Established H"older distortion bounds between different graphs

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.