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System of Inequalities in Continued Fractions from Finite Alphabets

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Abstract

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 \(\mathbf{A}\subseteq\mathbb{N}\).

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Authors and Affiliations

  1. Moscow Aviation Institute (National Research University), Moscow, 125993, Russia

    I. D. Kan & G. Kh. Solovyov

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  1. I. D. Kan
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  2. G. Kh. Solovyov
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Correspondence to I. D. Kan.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 197–206 https://doi.org/10.4213/mzm13580.

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Kan, I.D., Solovyov, G.K. System of Inequalities in Continued Fractions from Finite Alphabets. Math Notes 113, 212–219 (2023). https://doi.org/10.1134/S0001434623010248

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  • DOI: https://doi.org/10.1134/S0001434623010248

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