Geodesic
Linear Congruences in Continued Fractions on Finite Alphabets
Matematičeskie zametki, Tome 103 (2018) no. 6, pp. 853-862

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: linear congruence, continued fraction. 
@article{MZM_2018_103_6_a4,
     author = {I. D. Kan},
     title = {Linear {Congruences} in {Continued} {Fractions} on {Finite} {Alphabets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {853--862},
     publisher = {mathdoc},
     volume = {103},
     number = {6},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_6_a4/}
}
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I. D. Kan. Linear Congruences in Continued Fractions on Finite Alphabets. Matematičeskie zametki, Tome 103 (2018) no. 6, pp. 853-862. http://geodesic.mathdoc.fr/item/MZM_2018_103_6_a4/