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Dimensions of the popcorn graph

Haipeng Chen, Jonathan M. Fraser, Han Yu

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-Erdős inequality from probability theory.

Dimensions of the popcorn graph

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-Erdős inequality from probability theory.

Paper Structure

This paper contains 10 sections, 14 theorems, 70 equations, 2 figures.

Table of Contents

  1. The popcorn function
  2. Box dimension of the popcorn graph
  3. Assouad spectrum and Assouad dimension of the popcorn graph
  4. Preliminaries and notation
  5. Box dimensions: proof of Theorem \ref{['dim_B_result']}
  6. Preparation
  7. Proof of Theorem \ref{['dim_B_result']}
  8. Assouad spectrum: proof of Theorem \ref{['A_spe_result']}
  9. Preparation
  10. Proof of Theorem \ref{['A_spe_result']}

Key Result

Theorem 2.1

The box dimensions of the popcorn graph and full popcorn set are $4/3$, that is, $\dim_B F = \dim_B G_f = 4/3.$

Figures (2)

  • Figure 1: Popcorn graph
  • Figure 2: Full popcorn set

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.1
  • Definition 3.1
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 5.1: Estimate of Euler totient function
  • Theorem 5.2: Chung-Erdős Inequality
  • Theorem 5.3: Duffin-Schaeffer estimate
  • Lemma 5.1: Counting integers in horizontal strips
  • proof
  • ...and 15 more