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Survey article: on weighted densities and their connection with the first digit phenomenon. (English) Zbl 1237.60004
Summary: This paper is a general treatment of the various notions of densities used in papers on mantissa distribution of sequences of numbers. Equivalence classes of weighted densities are identified, and their hierarchy is stated. This permits us to give clear answers to several questions about the first digit phenomenon. Moreover, however light the weights are, we exhibit an example of a sequence of positive numbers whose mantissae do not admit any distribution in the sense of the corresponding density.

MSC:
60B10 Convergence of probability measures
11B05 Density, gaps, topology
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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