The dimension spectrum of the infinitely generated Apollonian gasket

TL;DR

This work proves that the dimension spectrum of the infinitely generated Apollonian gasket is full, i.e. $DS(\mathcal{A})=[0, dim_H(J_{\mathcal{A}})]$. It models the gasket as a limit set of an infinite conformal IFS and employs a computer-assisted, flexible framework within thermodynamic formalism to rigorously estimate Hausdorff dimensions of subsystems. A central contribution is an explicit distortion bound $K_{\mathcal{A}} \le 5.900319$ (with sharper bounds for subsystems) enabling a bootstrapping procedure to cover the spectrum across 18 steps. The methodology advances dimension-theory techniques for conformal fractals and provides a robust template for analyzing dimension spectra of gasket-type systems and similar limit sets.

Abstract

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors in [3] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed in [5]. Another key ingredient in our proof is obtaining reasonably sized distortion constants for the (infinite) Apollonian iterated function system.

Figures (5)

• Figure 1: 1 iteration
• Figure 2: 2 iterations
• Figure 3: Points on the circles
• Figure 4: A sequence of Apollonian circles
• Figure 5: The graph of $h: \mathbb R^+ \to \mathbb R^+$

Theorems & Definitions (19)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5