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It is known that if X is uniformly distributed modulo 1 and Y is an arbitraryrandom variable independent of X then Y+X is also uniformly distributed modulo1. We prove a converse for any continuous random variable Y (or a reasonableapproximation to a continuous random variable) so that if X and Y+X are equallydistributed modulo 1 and Y is independent of X then X is uniformly distributedmodulo 1 (or approximates the uniform distribution equally reasonably). Thistranslates into a characterization of Benford's law through a generalization ofscale-invariance: from multiplication by a constant to multiplication by anindependent random variable.  We also show a base-invariance characterization: if a positive continuousrandom variable has the same significand distribution for two bases then it isBenford for both bases. The set of bases for which a random variable is Benfordis characterized through characteristic functions.

Notes on scale-invariance and base-invariance for Benford's Law