Metric properties of continued fractions with large prime partial quotients
Wanjin Cheng, Wen Wu
TL;DR
This work advances the metric theory of continued fractions with two large prime partial quotients. It defines $E'(\phi)$ and proves a sharp zero-one Lebesgue measure law: $\mathcal L(E'(\phi))=0$ if $\sum_{n\ge1}\frac{n}{\phi(n)^2\log^2\phi(n)}<\infty$, and $\mathcal L(E'(\phi))=1$ if the sum diverges. The Hausdorff dimension of $E'(\phi)$ is determined via a pressure-function framework, with the critical value depending on $B=\liminf_{n\to\infty}\frac{\log\phi(n)}{n}$ and $b=\liminf_{n\to\infty}\frac{\log\log\phi(n)}{n}$, yielding explicit formulas in the regimes $1\le B<\infty$ and $B=\infty$. The analysis combines Cantor-set constructions, tailored mass distributions, and prime-distribution estimates to derive sharp dimension bounds, extending prior results for unrestricted large quotients and their prime-restricted variants. Overall, the paper deepens understanding of how simultaneous large prime partial quotients govern metric and fractal properties of continued-fraction expansions.
Abstract
Let $x \in [0,1)$ with continued fraction expansion $[a_1(x),a_2(x),\dots]$, and let $φ:\mathbb{N}\to\mathbb{R}^+$ 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 $a'_i(x)$ denotes $a_i(x)$ if $a_i(x)$ is prime and $0$ otherwise. We establish a zero-one law for the Lebesgue measure of $E'(φ)$ and determine its Hausdorff dimension.