A problem of Galambos on Engel expansions

Tom 92 / 2000

Jun Wu Acta Arithmetica 92 (2000), 383-386 DOI: 10.4064/aa-92-4-383-386

Streszczenie

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.

Autorzy

  • Jun Wu

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