Linear Inhomogeneous Congruences in Continued Fractions on Finite Alphabets

We consider the linear inhomogeneous congruence
$$ ax-by\equiv t\,(\operatorname{mod}q) $$
and prove an upper estimate for the number of its solutions. Here $a$, $b$, $t$, and $q$ are given natural numbers, $x$ and $y$ are coprime variables from a given interval such that the number $x/y$ expands in a continued fraction with partial quotients on a finite alphabet $\mathbf{A}\subseteq\mathbb{N}$. For $t=0$, a similar problem has been solved earlier by I. D. Kan and, for $\mathbf{A}=\mathbb{N}$, by N. M. Korobov. In addition, in one of the recent statements of the problem, an additional constraint in the form of a linear inequality was also imposed on the fraction $x/y$.