Linear Congruences in Continued Fractions on Finite Alphabets

A linear homogeneous congruence $ay\equiv bY \,(\operatorname{mod}{q})$ is considered and an order-sharp upper bound for the number of its solutions is proved. Here $a$, $b$, and $q$ are given jointly coprime numbers and $y$ and $Y$ are coprime variables in a given closed interval such that the number $y/Y$ can be expanded in a continued fraction with partial quotients from some alphabet $\mathbf{A}\subseteq\mathbb{N}$. For $\mathbf{A}=\mathbb{N}$ (and without the assumption that $y$ and $Y$ are coprime), a similar problem was solved by N. M. Korobov.