New Approximations for the Area of the Mandelbrot Set

TL;DR

This paper introduces new methods to approximate the Mandelbrot set's area using improved bounds, parallel algorithms, and 2-adic valuation analysis, advancing understanding of its fractal geometry.

Contribution

It provides novel upper bounds for the Mandelbrot set's area based on computational and number-theoretic techniques, improving previous estimates.

Findings

Derived improved upper bounds for the Mandelbrot set's area

Implemented a parallel computing algorithm for calculations

Analyzed 2-adic valuations of series coefficients

Abstract

Due to its fractal nature, much about the area of the Mandelbrot set $M$ remains to be understood. While a series formula has been derived by Ewing and Schober to calculate the area of $M$ by considering its complement inside the Riemann sphere, to date the exact value of this area remains unknown. This paper presents new improved upper bounds for the area based on a parallel computing algorithm and for the 2-adic valuation of the series coefficients in terms of the sum-of-digits function.