Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hausdorff dimension of the Apollonian gasket

Polina Vytnova, Caroline Wormell

Abstract

The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals.

Hausdorff dimension of the Apollonian gasket

Abstract

The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals.
Paper Structure (18 sections, 25 theorems, 140 equations, 1 figure, 1 table, 3 algorithms)

This paper contains 18 sections, 25 theorems, 140 equations, 1 figure, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Functional analysis setup
  3. Inducing
  4. Transfer operators and Ruelle--Bowen formula
  5. Chebyshev approximation in multiple dimensions
  6. Hardy spaces
  7. Chebyshev approximation in multiple dimensions
  8. Proof of Proposition \ref{['p:ProjectionNorms']}
  9. Min-max method
  10. The Euler--Maclaurin formula
  11. Remainder term $R(L,N)$
  12. Sum of derivatives term $D(L,N)$
  13. Integral term $I(N)$
  14. Numerical estimates for Apollonian IFS
  15. Pointwise estimates for the induced IFS
  16. ...and 3 more sections

Key Result

Theorem 1.1

The Hausdorff dimension of the Apollonian circle packing is

Figures (1)

  • Figure 1: Left: The unit disk and its images under iterations of $A_1$, $A_2$, $A_3$. The square $\square = \{ (x,y) \mid 0 \le x \le 0.5,\, -0.25 \le y \le 0.25\}$, where the induced system is defined is shown. Right: The square $\square$ and its images under the maps $f_n$ for $n = 1, \ldots, 20$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 46 more