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A new approach to the theory of Engel’s series. (English) Zbl 0232.10028
The paper is concerned with the representation of a real number $$x\in (0,1]$$ by its so called Engel’s series: $$x= \sum_1^\infty (q_1q_2\cdots q_n)^{-1}$$. Here $$(q_n)$$ is a nondecreasing sequence of integers $$\ge 2$$ constructed in the following way: $$q_1$$ is the least integer with $$q_1^{-1}<x$$, $$q_2$$ is the least integer with $$q_1^{-1}+ (q_1q_2)^{-1}<x$$ and so on. When $$x$$ is assumed to be uniformly distributed in $$[0,1]$$ one can regard the $$q_n$$’s as random variables. The paper studies the limiting behaviour of these random variables. Proofs of known results due to É. Borel [C. R. Acad. Sci., Paris 225, 773 (1947; Zbl 0029.15303)] and to P. Lévy [C. R. Acad. Sci., Paris 225, 918–919 (1947; Zbl 0029.15304)] are simplified considerably due to the observation that the random variables $$(\varepsilon_k)$$, $$(\varepsilon_k)$$ denoting the number of $$q_n$$’s equal to $$k$$, are independent.
Furthermore, it is proved (Theorem 6) that for almost all $$x$$ the sequence $$(q_n)$$ is strictly increasing for $$n\ge n_0(x)$$, where $$n_0(x)$$ depends on $$x$$. It is pointed out that some of the results remain valid if the distribution of $$x$$ is just assumed to be absolutely continuous on $$[0,1]$$. Finally, a modification of Engel’s series is suggested.
Reviewer: F. Topsøe

MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension