A problem of Galambos on Engel expansions

Jun Wu

Acta Arithmetica (2000)

  • Volume: 92, Issue: 4, page 383-386
  • ISSN: 0065-1036

Abstract

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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 ≥ 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) l i m 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. d i m H x ∈ ( 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.

How to cite

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Jun Wu. "A problem of Galambos on Engel expansions." Acta Arithmetica 92.4 (2000): 383-386. <http://eudml.org/doc/207394>.

@article{JunWu2000,
abstract = {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.},
author = {Jun Wu},
journal = {Acta Arithmetica},
keywords = {Engel expansions; Galambos's conjecture; Hausdorff dimension},
language = {eng},
number = {4},
pages = {383-386},
title = {A problem of Galambos on Engel expansions},
url = {http://eudml.org/doc/207394},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Jun Wu
TI - A problem of Galambos on Engel expansions
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 4
SP - 383
EP - 386
AB - 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.
LA - eng
KW - Engel expansions; Galambos's conjecture; Hausdorff dimension
UR - http://eudml.org/doc/207394
ER -

References

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  1. [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. Zbl0107.27002
  2. [2] K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 1990. 
  3. [3] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976. Zbl0322.10002
  4. [4] J. Galambos, The Hausdorff dimension of sets related to g-expansions, Acta Arith. 20 (1972), 385-392. Zbl0213.06701
  5. [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. Zbl0198.38104

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