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## Acta Arithmetica

2000 | 92 | 4 | 383-386
Tytuł artykułu

### A problem of Galambos on Engel expansions

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is,
(1) $x =1/d₁(x) + 1/(d₁(x)d₂(x)) + ... + 1/(d₁(x)d₂(x)...d_n(x)) + ...$,
where ${d_{j}(x), j ≥ 1}$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}(x) ≥ d_{j}(x)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1],
(2) $lim_{n→∞} d_{n}^{1/n}(x) =e. He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem.$dim_H{x ∈ (0,1]: (2) fails} = 1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and$dim_{H}\$ to denote the Hausdorff dimension.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
383-386
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-06-10
Twórcy
autor
• Department of Mathematics and Center of Non-linear Science, Wuhan University, 430072, Wuhan, People's Republic of China
Bibliografia
• [1] P. Erdős, A. Rényi and P. Szüsz, On Engel's and Sylvester's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 7-32.
• [2] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 1990.
• [3] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976.
• [4] J. Galambos, The Hausdorff dimension of sets related to g-expansions, Acta Arith. 20 (1972), 385-392.
• [5] J. Galambos, The ergodic properties of the denominators in the Oppenheim expansion of real numbers into infinite series of rationals, Quart. J. Math. Oxford Ser. (2) 21 (1970), 177-191.
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