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Proceedings of the American Mathematical Society

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The rate of growth of the denominators in the Oppenheim series


Author: János Galambos
Journal: Proc. Amer. Math. Soc. 59 (1976), 9-13
MSC: Primary 10K10
DOI: https://doi.org/10.1090/S0002-9939-1976-0568142-4
MathSciNet review: 0568142
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Abstract: A Borel-Cantelli lemma is proved for a sequence of functions of the denominators in the Oppenheim expansion of real numbers. This is then applied to the study of the rate of growth of the denominators in the above series. The laws obtained are almost sure type, that is, valid for (Lebesgue) almost all $ x$ in the unit interval. The results are new even for the classical expansions of Engel, Sylvester and Cantor (product).


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DOI: https://doi.org/10.1090/S0002-9939-1976-0568142-4
Keywords: Oppenheim series, denominators, Borel-Cantelli lemma, asymptotic laws, $ \lim \;\sup $, $ \lim \;\inf $, Engel series, Sylvester series, Cantor product
Article copyright: © Copyright 1976 American Mathematical Society