The rate of growth of the denominators in the Oppenheim series

Galambos, János

doi:10.1090/S0002-9939-1976-0568142-4

The rate of growth of the denominators in the Oppenheim series
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Proc. Amer. Math. Soc. 59 (1976), 9-13 Request permission

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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