### A problem of Galambos on Engel expansions

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d\u2081\left(x\right)+1/\left(d\u2081\left(x\right)d\u2082\left(x\right)\right)+...+1/(d\u2081\left(x\right)d\u2082\left(x\right)...{d}_{n}\left(x\right))+...$, where ${d}_{j}\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and ${d}_{j+1}\left(x\right)\ge {d}_{j}\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty}{d}_{n}^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_{H}$ to denote the Hausdorff...