For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, and $\alpha_{i+1} = \frac{1}{\alpha_i - a_i}$ for $i \geq 0$. A convergent of $x$ is a rational number $p_n/q_n$ where $p_n/q_n = [a_0; a_1, \cdots, a_n]$, and $\gcd(p_n, q_n) = 1$.

My question is as follows. Is there a set $\mathcal{A} \subset \mathcal{P}$, where $\mathcal{P}$ denotes the set of prime numbers, and the set of $\mathfrak{C}_\mathcal{A}$ real numbers $x \in [0,1]$ with $x = [0; a_1, a_2, \cdots]$ such that $a_i \in \mathcal{A}$ for all $i \geq 1$, with the property that there exist infinitely many pairs of primes $p < q$ such that $p/q$ is a convergent for some element $x \in \mathfrak{C}_\mathcal{A}$?

The question is of interest because of the following 'additive' property of continued fractions:

$$\displaystyle \frac{1}{a + \frac{b}{d}} = \frac{d}{b + ad}$$

which implies that if $p/q$ is a convergent in $\mathfrak{C}_\mathcal{A}$ then so is $q/(p + aq)$. This is interesting because Zaremba's conjecture asserts that for a finite set of the form $\mathcal{B} = \{1, 2, \cdots, B\}$ with $B \geq 5$, the set of denominators $\mathfrak{D}_\mathcal{B}$ of those $d$ that appears as the denominator of a fraction $b/d$ which is the convergent of some number $z$ whose partial quotients all line $\mathcal{B}$ should be all but finitely many positive integers. If this hypothesis can be relaxed to a set of primes, then one can show that almost all positive integers can be written as the sum $p + aq$ where $a$ is from a finite family of primes which would greatly strengthen Chen's theorem and be ever so close to the vaunted Goldbach conjecture.