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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.

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