Metric properties of continued fractions with large prime partial quotients

Let $x\in [0,1)$ with continued fraction expansion $[a_1(x),a_2(x),\dots ]$, and let $\phi :\mathbb{N}\to {\mathbb{R}}^{+}$ be a non-decreasing function. We consider the numbers whose continued fraction expansions contain at least two partial quotients that are simultaneously large and prime, that is $$E^{\mathrm{\prime }}(\phi ):=\{x\in [0,1):\mathrm{\exists }1\le k\mathrm{\ne }l\le n,a_k^{\mathrm{\prime }}(x),a_l^{\mathrm{\prime }}(x)\ge \phi (n)\text{ for i.m. }n\in \mathbb{N}\},$$ where $a_i^{\mathrm{\prime }}(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E^{\mathrm{\prime }}(\phi )$ and determine its Hausdorff dimension.