Geodesic
Linear Inhomogeneous Congruences in Continued Fractions on Finite Alphabets
Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 412-425

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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 $x/y$.
Keywords: linear inhomogeneous congruence, linear homogeneous congruence, continued fraction, finite alphabet. 
@article{MZM_2022_112_3_a9,
     author = {I. D. Kan and V. A. Odnorob},
     title = {Linear {Inhomogeneous} {Congruences} in {Continued} {Fractions} on {Finite} {Alphabets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {412--425},
     publisher = {mathdoc},
     volume = {112},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a9/}
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I. D. Kan; V. A. Odnorob. Linear Inhomogeneous Congruences in Continued Fractions on Finite Alphabets. Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 412-425. http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a9/