| Abstract: |
| 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. |
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