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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0568142-4
PII: S 0002-9939(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