On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set
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 intu...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/10/534 |
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| author | Filippo Beretta Jesse Dimino Weike Fang Thomas C. Martinez Steven J. Miller Daniel Stoll |
| author_facet | Filippo Beretta Jesse Dimino Weike Fang Thomas C. Martinez Steven J. Miller Daniel Stoll |
| author_sort | Filippo Beretta |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T20:13:08Z |
| format | Article |
| id | doaj.art-f53bae9934df49aeb71c926b7bcb7952 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T20:13:08Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-f53bae9934df49aeb71c926b7bcb79522023-11-24T00:11:06ZengMDPI AGFractal and Fractional2504-31102022-09-0161053410.3390/fractalfract6100534On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot SetFilippo Beretta0Jesse Dimino1Weike Fang2Thomas C. Martinez3Steven J. Miller4Daniel Stoll5Department of Mathematics, University of Milan, 20133 Milan, ItalyDepartment of Mathematics, CUNY Graduate Center, New York, NY 10016, USADepartment of Mathematics, University of Notre Dame, South Bend, IN 46556, USADepartment of Mathematics, Harvey Mudd College, Claremont, CA 90701, USADepartment of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USADepartment of Mathematics, University of Michigan, Ann Arbor, MI 48109, USAWe 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.https://www.mdpi.com/2504-3110/6/10/534complex dynamicsbenford’s lawpower series |
| spellingShingle | Filippo Beretta Jesse Dimino Weike Fang Thomas C. Martinez Steven J. Miller Daniel Stoll On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set Fractal and Fractional complex dynamics benford’s law power series |
| title | On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set |
| title_full | On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set |
| title_fullStr | On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set |
| title_full_unstemmed | On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set |
| title_short | On Benford’s Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set |
| title_sort | on benford s law and the coefficients of the riemann mapping function for the exterior of the mandelbrot set |
| topic | complex dynamics benford’s law power series |
| url | https://www.mdpi.com/2504-3110/6/10/534 |
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