Restricted slowly growing digits for infinite iterated function systems

Abstract

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

14 pages, 2 figures