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The significant-digit phenomenon. (English) Zbl 0833.60003
The paper is a survey article on scale-invariance and base-invariance. In order to explain the significant-digit phenomenon (Benford’s law) the author defines a natural domain \(\mathcal A\) on the positive reals which is generated by infinite basis sets \(S_{a,b} = \bigcup^\infty_{i = - \infty} [a 10^i, b 10^i)\), \(1 \leq a < b \leq 10\). On \(\mathcal A\) significant-digit laws to base 10 can be studied. A probability measure \(P\) is called scale-invariant if (i) \(P(S) = P(\alpha S)\) for all real \(\alpha > 0\) and all \(S \in {\mathcal A}\). A probability measure \(P\) is called base-invariant if (ii) \(P(S_{1,b}) = P(S^{1/k}_{1,b})\) for all positive integers \(k\) and all basis sets \(S_{1,b}\), where \(S^{1/k} = \{x^{1/k} : x \in S\}\) [cf. the author, Proc. Am. Math. Soc. 123, No. 3, 887-895 (1995; Zbl 0813.60002)]. Both scale-invariance and base-invariance imply Benford’s law, essentially. The special role of the constant 1 is discussed.
Reviewer’s remarks: 1. Under base-invariance of a probability measure \(p\) the defining relation (ii) holds for all \(S \in {\mathcal A}\), not only for basis sets. 2. For the definition of scale-invariance of a probability measure \(P\) it suffices that the defining relation (i) is satisfied only for basis sets \(S_{1,b}\). Then it holds for all \(S \in {\mathcal A}\). After this, the definitions and properties of scale-invariance and base- invariance would look more alike. 3. The construction of \(\mathcal A\) has the advantage of avoiding random variables in the definition of scale- invariance and base-invariance but the disadvantage of not explaining the simultaneous validity of Benford’s law to several bases.

60A10 Probabilistic measure theory
60E99 Distribution theory
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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