论文标题
关于本福德的法律和Riemann映射功能的系数
On Benford's Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set
论文作者
论文摘要
我们研究本福德与分形几何形状有关的定律。基本的分形（例如Cantor Set和Sierpinski三角形）作为迭代集的极限获得，其组件的独特度量遵循几何分布，这在大多数基础上都是Benford。在这种直觉的基础上，我们旨在研究更复杂的分形中的这种分布。我们从设置为单位磁盘的补体的曼德尔布罗特（Mandelbrot）的外部检查了riemann映射的laurent系数和其相互函数的泰勒系数。这些系数是2个ADIC合理的数字，通过统计测试，我们证明了分子和分母非常适合Benford定律。我们提供了有关这些系数的其他猜想和观察。特别是，我们重点介绍了与系数的分母有关的某些算术子序列，对其斜率进行了估计，并描述了计算它们的有效方法。
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.
