Zaremba's conjecture asserts that if there exists some integer $A > 1$ such that for the set of numbers $x \in [0,1]$ whose partial quotients in the continued fraction representation are all bounded from above by $A$ contains all natural numbers as partial denominators. In other words, for each $d \in \mathbb{N}$ there exists $x$ with all partial quotients $0 = a_0, a_1, \cdots \leq A$ and an integer $n$ such that $[0; a_1, \cdots, a_n] = \frac{b}{d}$. Bourgain and Konotorvich recently proved that Zaremba's conjecture holds for a set of density 1, which may depend on the choice of $A$.
Is the following easier version of Zaremba's conjecture trivial? That is, if we take our partial quotients from a set $\mathcal{A}$ which is an infinite but sparse set, that one should immediately obtain all possible $d \in \mathbb{N}$ as partial denominators?