Hausdorff dimensions in Engel expansions (Q2717768)

The authors consider the Engel expansion of points \(x\) in the interval \((0,1]\). The Engel expansion is defined as NEWLINE\[NEWLINEx={1\over d_1(x)}+ {1\over d_1(x) d_2(x)}+ \cdots+ {1\over d_1(x)d_2(x) \cdots d_n(x)}+ \cdots,NEWLINE\]NEWLINE where \(\{d_j(x),j\geq 1\}\) is a sequence of positive integers satisfying \(d_1(x) \geq 2\) and \(d_{j+1}(x)\geq d_j(x)\) for \(j\geq 1\). In [Representations of real numbers by infinite series, Lecture Notes Math. 502, Springer, Berlin (1976; Zbl 0322.10002)], \textit{J. Galambos} proved that \(\lim_{n\to \infty}d_n^{1/n} (x)=e\) for almost all \(x\in[0,1)\), and posed questions concerning the Hausdorff dimension of the set of points where this limit fails. In this paper, the authors show that the Hausdorff dimension of the set of points \(A (\alpha)\) for which \(\lim_{n\to\infty} d_n^{1/n}(x)=\alpha\) is 1 for each \(\alpha\geq 1\). They obtain their results by constructing a limsup subset contained in \(A(\alpha)\) and using the mass distribution principle.