Dimensions of the popcorn graph

TL;DR

This paper determines the fractal dimensions of the popcorn function's graph, revealing its complex geometric structure using advanced mathematical tools from analysis, number theory, and probability.

Contribution

It computes the box, Assouad, and spectrum dimensions of the popcorn function's graph, providing new insights into its fractal properties.

Findings

Box dimension of the graph is 4/3

Computed the Assouad dimension and spectrum

Used Diophantine approximation and probability theory techniques

Abstract

The 'popcorn function' isThe `popcorn function' is a well-known and important example in real analysis with many interesting features. We prove that the box dimension of the graph of the popcorn function is 4/3, as well as computing the Assouad dimension and Assouad spectrum. The main ingredients include Duffin-Schaeffer type estimates from Diophantine approximation and the Chung-Erd\H{o}s inequality from probability theory.