Hausdorff dimension of sets with restricted, slowly growing partial quotients in the semi-regular continued fraction

We consider sets of irrational numbers whose partial quotients $a_{\sigma,n}$ in the semi-regular continued fraction expansion obey certain restrictions and growth conditions. Our main result asserts that, for any sequence $\sigma\in\{-1,1\}^\mathbb N$ in the expansion, any infinite subset $B$ of $\mathbb N$ and for any function $f$ on $\mathbb N$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrationals in $(0,1)$ such that \[ a_{\sigma,n}\in B,\ a_{\sigma,n}\leq f(n)\text{ for all $n\in\mathbb N$ and }a_{\sigma,n}\to\infty\text{ as }n\to\infty\] is of Hausdorff dimension $\tau(B)/2,$ where $\tau(B)$ is the exponent of convergence of $B$. We also prove that for any $\sigma\in\{-1,1\}^\mathbb N$ and any $B\subset\mathbb N$, the set of irrationals in $(0,1)$ such that \[ a_{\sigma,n}\in B\text{ for all $n\in\mathbb N$ and }a_{\sigma,n}\to\infty\text{ as }n\to\infty\] is also of Hausdorff dimension $\tau(B)/2$. To prove these results, we construct non-autonomous iterated function systems well-adapted to the given restrictions and growth conditions, and then apply the dimension theory developed by Rempe-Gillen and Urba\'nski.Comment: 14 pages, no figur