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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 d x/x and (2) the scale invariance of d x/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.

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Acknowledgements

The second named author was supported by NSF Grant DMS1265673. We are grateful to the referees for numerous comments which greatly improved the exposition of the results.

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Correspondence to Steven J. Miller.

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Manack, C., Miller, S.J. Leading digit laws on linear Lie groups. Res. number theory 1, 22 (2015). https://doi.org/10.1007/s40993-015-0024-4

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  • DOI: https://doi.org/10.1007/s40993-015-0024-4

Keywords

  • Benford’s law
  • Digit laws
  • Haar measure
  • Matrix groups

Mathematics subject classification

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