Geodesic
System of Inequalities in Continued Fractions from Finite Alphabets
Matematičeskie zametki, Tome 113 (2023) no. 2, pp. 197-206

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The system of two inequalities $$ \biggl|\frac yx-\psi_1\biggr|\le \varepsilon_1\qquad \text{and} \qquad \biggl\|\frac{ay}x-\psi_2\biggr\|\le \varepsilon_2 $$ is considered, and an upper bound for the number of its solutions is established. Here $a$, $\psi_1$, $\psi_2$, $\varepsilon_1$, and $\varepsilon_2$ are given real numbers, $\varepsilon_1$ and $\varepsilon_1$ are positive and arbitrarily small, $\|\cdot\|$ is the distance to the nearest integer, and $x$ and $y$ are coprime variables from given intervals such that the partial quotients of the continued fraction expansion of $y/x$ belong to a finite alphabet $\mathbf{A}\subseteq\mathbb{N}$.
Keywords: inequality, distance to the nearest integer, continued fraction, finite alphabet. 
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     author = {I. D. Kan and G. Kh. Solov'ev},
     title = {System of {Inequalities} in {Continued} {Fractions} from {Finite} {Alphabets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {197--206},
     publisher = {mathdoc},
     volume = {113},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_113_2_a3/}
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I. D. Kan; G. Kh. Solov'ev. System of Inequalities in Continued Fractions from Finite Alphabets. Matematičeskie zametki, Tome 113 (2023) no. 2, pp. 197-206. http://geodesic.mathdoc.fr/item/MZM_2023_113_2_a3/