Leading Digit Laws on Linear Lie Groups

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 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 logB(s); thus the first digit is d with probability logB(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.

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