A problem of Galambos on Engel expansions

• Autorzy: Jun Wu
• 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.

Kategorie tematyczne

• 28A80: Fractals
• 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2000
• Tom: 92
• Numer: 4
• Strony: 383-386

Daty

wydano

2000

otrzymano

1999-06-10

Twórcy

autor

Jun Wu

• 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.