On Benford's Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set

TL;DR

This paper explores the occurrence of Benford's law in the coefficients of Riemann mappings related to fractals, especially the Mandelbrot set, revealing that these coefficients' numerators and denominators follow Benford's distribution.

Contribution

It extends the study of Benford's law to complex fractal coefficients, providing statistical evidence, conjectures, and computational methods for these coefficients in the context of the Mandelbrot set.

Findings

Coefficients' numerators and denominators fit Benford's law.

Identified arithmetic subsequences related to coefficients.

Provided estimates and efficient computation methods.

Abstract

We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford's law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients' denominators, provide an estimate for their slope, and describe efficient methods to compute them.