The first digit property for exponential sequences is independent of the underlying distribution.

*(English)*Zbl 0608.10054The remarkable empirical conjecture known as Benford’s law has elicited a great deal of interesting research, in search for theoretical justification. F. Benford [Proc. Am. Philos. Soc. 78, 551-572 (1938; Zbl 0018.26502)] noted that the proportion of entries in random tables of data whose first significant non-zero digit is n, where \(1\leq n\leq 9\), tended towards \(\log_{10}((n+1)/n)\), instead of 1/9, as might have been expected. Thus the leading digit 1 appears about 3/10 of the time, while 9 occurs less than 1/20 of the time. This phenomenon is commonly called Benford’s law, even though S. Newcomb, an astronomer and at one time the editor of the American Journal of Mathematics, had conjectured the law 57 years earlier [Am. J. Math. 4, 39-41 (1881)]. For literature surveys, see D. E. Knuth [The art of computer programming, Vol. 2 (Addison-Wesley 1969; Zbl 0191.180)] and R. A. Raimi [Am. Math. Mon. 83, 521-538 (1976; Zbl 0349.60014)].

Using tools from the theory of uniform distribution mod 1, P. Diaconis [Ann. Prob. 5, 72-81 (1977; Zbl 0364.10025)] has established Benford’s law for a large class of arithmetic sequences, including the sequence \(\{cr^ n\); \(n=1,2,...\}\) with \(c>0\) and \(r>1\), where \(\log_{10}r\) is irrational. The natural density in the sequence \(\{cr^ n\}\) of elements beginning with the first digit \(\ell\) is \(\log_{10}((\ell +1)/\ell)\). The authors of the paper under review prove that this Benford property persists for any finitely additive, translation invariant density on sequences of the form \(\{cr^ n+a_ n\); \(n=1,2,...\}\) with \(c>0\), where \(\log_{10}r\) is irrational and \(a_ n=o(r^ n)\). This includes the Fibonacci numbers, where \(c=1/\sqrt{5}\), \(r=(1+\sqrt{5})/2\) and \(a_ n=-(1/5)((1- \sqrt{5})/2)^ n\).

Using tools from the theory of uniform distribution mod 1, P. Diaconis [Ann. Prob. 5, 72-81 (1977; Zbl 0364.10025)] has established Benford’s law for a large class of arithmetic sequences, including the sequence \(\{cr^ n\); \(n=1,2,...\}\) with \(c>0\) and \(r>1\), where \(\log_{10}r\) is irrational. The natural density in the sequence \(\{cr^ n\}\) of elements beginning with the first digit \(\ell\) is \(\log_{10}((\ell +1)/\ell)\). The authors of the paper under review prove that this Benford property persists for any finitely additive, translation invariant density on sequences of the form \(\{cr^ n+a_ n\); \(n=1,2,...\}\) with \(c>0\), where \(\log_{10}r\) is irrational and \(a_ n=o(r^ n)\). This includes the Fibonacci numbers, where \(c=1/\sqrt{5}\), \(r=(1+\sqrt{5})/2\) and \(a_ n=-(1/5)((1- \sqrt{5})/2)^ n\).

Reviewer: O.P.Stackelberg