Metric properties of continued fractions with large prime partial quotients

Abstract

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 \[ E'(\phi):=\Big\{x\in[0,1): \exists\, 1\leq k\neq l\leq n, \ a'_{k}(x),\ a'_{l}(x)\geq\phi(n) \ \text{for i.m. } n\in\mathbb{N}\Big\}, \] where $a'_i(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E'(\phi)$ and determine its Hausdorff dimension.