Leading Digit Laws on Linear Lie Groups

TL;DR

This paper explores the distribution of leading digits in matrix components of linear Lie groups, generalizing Benford's law through Haar measure invariance and analyzing digit laws on spheres with high-dimensional limits.

Contribution

It extends Benford's law to matrix components of linear Lie groups using Haar measure invariance and investigates digit distributions on spheres with increasing dimension.

Findings

Leading digit laws follow from Haar measure invariance on Lie groups.

Distribution of digits on spheres exhibits periodic behavior in high dimensions.

Generalization of Benford's law to matrix components of Lie groups.

Abstract

We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.