Polymath: Benford's Law and Fractal Geometry

Introduction

Calculating the coefficients of the Riemann mapping function for the exterior of the Mandelbrot set to the closed unit disk.

Compute Coefficients

To compute am and bm coefficients of the Laurent series of the Riemann mapping function, use codes/coeff_compute.py

For example, to compute the first 7168 coefficients for both am and bm

$ python codes/coeff_compute.py --max_power 7168

Results will be saved in the same folder with names "am_coeffs_7168.csv" and "bm_coeffs_7168.csv." Intermediate results will also be saved for every 1024 coefficients.

If existing am and/or bm coefficients are computed, we can load coefficients from existing files (say bm_coeffs_1024.csv and am_coeffs_1024.csv)

$ python codes/coeff_compute.py --max_power 7168 --load_bm "bm_coeffs_1024.csv" --load_am "am_coeffs_1024.csv"

Sampling simulation

To conduct simulations to see sampling error, use Data Analysis/sampling_simulation.ipynb

Test Statistics and Power Computation

We provide two jupyter notebooks for computing chi-square test statistics and draw all plots in our work. They are named as amChiSquareData.ipynb and bmChiSquareData.ipynb.

For results related to log base 10 modulo 1, please refer to amLogData.ipynb and bmLogData.ipynb.

To compute the power of each tests, we provide a Excel sheet and instructions to compute the power inside the sheet. There are also equivalent chi-square test statistics for cross-checking.

For example, to compute the power of a test with 0.05 significance level with degree of freedom equals 8, we can use the following R command (lambda should be substituted with the value automatically computed on the Excel sheet).

$ 1 - qchisq(.95, 8), 8, lambda)

To conduct simulations to see sampling error, use Data Analysis/sampling_simulation.ipynb