Research in Number Theory

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Leading digit laws on linear Lie groups

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Abstract

We study the leading digit laws for the matrix entries of a linear Lie group G. For non-compact G, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group \(\mathbb {R}^{+}\) is dx/x and (2) the scale invariance of dx/x implies the distribution of the digits follow Benford’s law. Viewing this scale invariance as left invariance of Haar measure, we see either Benford or power law behavior in the significands from one matrix entry of various such G. When G is compact, the leading digit laws we obtain come as a consequence of digit laws for a fixed number of components of a unit sphere. The sequence of digit laws for the unit sphere exhibits periodic behavior as the dimension tends to infinity.

Keywords

Benford’s law Digit laws Haar measure Matrix groups 

Mathematics subject classification

11K06 60F99 (primary) 28C10 15B52 15B99 (secondary) 

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© The Author(s) 2015

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsFranklin & MarshallLancasterUSA
  2. 2.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA

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