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

15 pages