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Remark on a paper by R. L. Duncan concerning the uniform distribution mod 1 of the sequence of the logarithms of the Fibonacci numbers. (English) Zbl 0212.39502
Fibonacci Q. 7, No. 5, 465-466, 473 (1969).
Let \((\mu_n)\) be the sequence of Fibonacci numbers defined by \(\mu_1=1\), \(\mu_2=2\) and \(\mu_n=\mu_{n-1}+\mu_{n-2}\) \((n\geq 3\). R. L. Duncan [Fibonacci Q. 5, 137–140 (1967; Zbl 0212.39501)] has shown that the sequence \((\log \mu_n)\) is uniformly distributed (mod 1), where the logarithm is taken to the base 10. In the paper under review the author gives a short proof of this result of Duncan, using the fact that if \(\xi=(1+\sqrt 5)/2\) and if \(\omega\neq 0\) is real and algebraic, then \(\xi^\omega\) is not an algebraic number, and appealing to a theorem due to J. G. van der Corput [Acta Math. 56, 373–456 (1931; Zbl 0001.20102)]. As a result, the sequence \((\log \mu_n)\) is uniformly distributed (mod 1), provided that the base of the logarithm is chosen in such a way that \(\log\xi\) is an irrational number. The author also proves that the sequence of integers \([(\log \mu_n)]\) is uniformly distributed in the sense of I. Niven [Trans. Am. Math. Soc. 98, 52–61 (1961; Zbl 0096.03102)].

MSC:
11J71 Distribution modulo one
11K06 General theory of distribution modulo \(1\)
11K31 Special sequences
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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