Mandelbrot set area via the Böttcher series

Let $\mathbb{C}$ be the complex plane, $M$ the Mandelbrot set, and $D$ the closed unit disk. There is an analytic Böttcher map

$$\phi : \mathbb{C} - D \to \mathbb{C} - M$$

$$\phi(z) = z + \sum_n b_n z^{-n}$$

and the area of the Mandelbrot set is

$$\mu(M) = \pi \left(1 - \sum_n n b_n^2\right)$$

Bittner et al. 2014 computed 5M terms of this series, resulting in the bound

$$\mu(M) \le 1.68288$$

We can compute out to $2^{27} = 134,217,728$ terms in a couple hours on an A100, producing

$$\mu(M) \le 1.651587035834859$$

We use expansion arithmetic, representing numbers as unevaluated sums of double precision numbers, as computing in double runs out of precision around $2^{23}$ terms.

Alas, the series approach isn't the fastest

The fastest of the mostly trustworthy methods I've seen for Mandelbrot area is Fisher and Hill 1993, who use the Koebe 1/4 theorem to prove (up to double precision) that quadree cells are either entirely outside or entirely inside the set. Their bounds are

$$1.50296686 < \mu(M) < 1.57012937$$

Worse, accurate extrapolation of the series seems out of reach: out to $2^{26}$ terms the area contribution locally averages to a noisy power law fit with exponent -1.08, but as shown by Bielefeld et al. 1988 this would violate non-Hölder continuity at the boundary. Attempts at extrapolating from our results out to infinity produce area estimates around 1.59 or 1.60, which is already contradicted by Fisher and Hill.

Explanation, analysis, and downloads

This colab has more explanation of the methods used, and some plots of the computed series. For example, here is a plot of the locally averaged area contributions vs. the illusory -1.08 power law fit:

For an .npy file with the series coefficients out to $2^k$ terms, set $k to 1, 2, ..., 26, or 27 and download

• https://storage.googleapis.com/mandelbrot/numpy/f-k$k.npy

The shape will be [2^k, 2] corresponding to expansion arithmetic with 2 doubles; sum across the last axis if you want a single double.

Building

We use the Meson build system, and the excellent high precision arithmetic library Arb for bootstrapping. We also depend on clang even when compiling CUDA, to allow more recent C++ features. To install dependencies:

# On Mac
brew install meson arb cmake pkg-config openssl

# On Debian Buster
echo deb http://apt.llvm.org/buster/ llvm-toolchain-buster-13 main | sudo tee -a /etc/apt/sources.list
echo deb-src http://apt.llvm.org/buster/ llvm-toolchain-buster-13 main | sudo tee -a /etc/apt/sources.list
sudo apt-get install clang-13 libc++-13-dev libc++abi-13-dev libomp-13-dev \
    python3 python3-pip python3-setuptools python3-wheel ninja-build \
    libmpfr-dev libflint-dev libflint-arb-dev libssl-dev
pip3 install --user meson

Then build and test with

./setup
cd build/release  # Or build/debug
meson compile
meson test

For CUDA profiling, use

# https://developer.nvidia.com/nsight-systems
# https://docs.nvidia.com/nsight-systems/UserGuide/index.html#example-single-command-lines
nsys profile --stats=true ./build/release/area_cuda_test