Modular Generalization of the Bourgain–Kontorovich Theorem

The set $\mathfrak{D}^N_\mathbf{A}$ of all irreducible denominators $\le N$ of positive rationals $<1$ whose continued fraction expansions consist of elements of the set $\mathbf{A}=\{1,2,4\}$ is considered. We prove that, for any prime $Q\le N^{2/3}$, the set $\mathfrak{D}^N_{\mathbf{A}}$ contains almost all possible remainders on division by $Q$ and the remainder term in the corresponding asymptotic formula decays according to a power law.