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For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with anattractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq\mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in\Lambda(\mathbf{f}): a_n(x)\in D \text{ for all }n\in\mathbb N \text{ and}\lim_{n\to\infty} a_n=\infty\}. \] For a function $\varphi:\mathbb{N}\to [\minD, \infty)$ such that $\varphi(n)\to\infty$ as $n\to\infty$, we compute theHausdorff dimension of the set $$ S(\mathbf{f},D,\varphi) = \left\{ x\in\E(\mathbf{f},D) : a_n(x)\leq \varphi(n) \text{ for all } n\in\mathbb N\right\}. $$ We prove that the Hausdorff dimension stays the same no matter howslowly the function $\varphi$ grows. One of the consequences of our result isthe recent work of Takahasi (2023), which only dealt with regular continuedfraction expansions. We further extend our result to slowly growing products of(not necessarily consecutive) digits.

Restricted slowly growing digits for infinite iterated function systems