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) ${\displaystyle x=\frac{1}{d}₁\left(x\right)+\frac{1}{d₁\left(x\right)d₂\left(x\right)}+...+\frac{1}{d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right)}+...}$, where ${\displaystyle \{d_j\left(x\right),j\ge 1\}}$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and ${\displaystyle d_{j+1}\left(x\right)\ge d_j\left(x\right)}$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) ${\displaystyle \underset{n\to \infty }{lim}{d}_n^{\frac{1}{n}}\left(x\right)=e.Heconjectured(\left[3\right],P132)t\hat{t}heHausd\text{or}ffdimensionofthesetwhere\left(2\right)failsiso\ne .Inthispaper,weprovethisconjecture:The\text{or}em.}$dim_H{x ∈ (0,1]: (2) fails} = 1${\displaystyle .WeuseL¹\to de\neg etheo\ne -dimensionalLebesguemeasureon(0,1]\text{and}}$dim_{H}!$! to denote the Hausdorff dimension.