A problem of Galambos on Engel expansions

Jun Wu

Jun Wu

Acta Arithmetica, Tome 92 (2000), p. 383-386 / Harvested from The Polish Digital Mathematics Library

Résumé

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x = 1 / d ₁ (x) + 1 / (d ₁ (x) d ₂ (x)) + . . . + 1 / (d ₁ (x) d ₂ (x) . . . d_{n} (x)) + . . .$ , where $d_{j} (x), j \geq 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j + 1} (x) \geq d_{j} (x)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $l i m_{n \to \infty} d_{n}^{1 / n} (x) = e .$ He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $d i m_{H} x \in (0, 1] : (2) f a i l s = 1$ . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $d i m_{H}$ to denote the Hausdorff dimension.

Publié le : 2000-01-01

Zbl 0949.11037

EUDML-ID : urn:eudml:doc:207394

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     author = {Jun Wu},
     title = {A problem of Galambos on Engel expansions},
     journal = {Acta Arithmetica},
     volume = {92},
     year = {2000},
     pages = {383-386},
     zbl = {0949.11037},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav92i4p383bwm}
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Jun Wu. A problem of Galambos on Engel expansions. Acta Arithmetica, Tome 92 (2000) pp. 383-386. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav92i4p383bwm/

Bibliographie

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[00001] [2] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 1990.

[00002] [3] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976. | Zbl 0322.10002

[00003] [4] J. Galambos, The Hausdorff dimension of sets related to g-expansions, Acta Arith. 20 (1972), 385-392. | Zbl 0213.06701

[00004] [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. | Zbl 0198.38104