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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

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