A simple proof for the upper bound of a theorem of T. Łuczak (Q6614510)

Given real numbers \(b,c >1\) and denoting by \(x = [0:a_1(x),a_2(x),\ldots]\) the classical continued fractions expansion of irrationals \(x \in [0,1]\), \textit{T. Łuczak} [Mathematika 44, No. 1, 50--53 (1997; Zbl 0884.11028)] proved that the Hausdorff dimension of\N\[\NE(b,c) = \{x \in [0,1): a_n(x) \geq c^{b^n} \text{ for infinitely many } n\}\N\]\Nis bounded above from \(1/(b+1)\). Since this matches the straight-forwardly obtained lower bound by Feng, Wu, Liang and Tseng [\textit{D.-J. Feng} et al., Mathematika 44, No. 1, 54--55 (1997; Zbl 0884.11029)] for\N\[\N\tilde{E}(b,c) = \{x \in [0,1): a_n(x) \geq c^{b^n} \text{ for all } n\},\N\]\Nand clearly \(\tilde{E}(b,c) \subseteq E(b,c)\), this proved the exact Hausdorff dimension of both sets.\N\NAlthough Łuczak's proof is quite short, it contained some non-trivial covering arguments. In this article, the authors provide an even shorter and simpler proof of Łuczak's result that comes without any covering argument at all. In short, the idea is to observe (which was essentially already a step in the proof of Łuczak) that for any \(1/b < s < 1\) and \(1/s < d < b\) and any \(x \in E(b,c)\) we obtain infinitely many \(n\) \[\tag{1}a_{n+1}(x) > \left(a_1(x) \cdots a_n(x)^{sd-1}c^{(1-s)d^{n+1}}\right).\]\N\NThe authors now show that the set of all \(x\) satisfying $(1)$ (after \(s \to 1\) and \(d \to b\)) has Hausdorff dimension at most \(1/(b+1)\), which clearly implies the claim. After writing the set that satisfies $(1)$ as a limsup set, the authors estimate the Lebesgue measure of cylinder sets \(I(b_1,\ldots,b_n) :=\{x \in [0,1]: a_1(x) = b_1,\ldots, a_n(x) = b_n\}\) and the ratios of \(\frac{\vert I(b_1,\ldots,b_n)\vert}{\vert I(b_1,\ldots,b_n,j)\vert}\) (which are both well-studied, classical estimates in the metric theory of continued fractions) to obtain the desired result.