On a case of joint Dirichlet approximations
https://doi.org/10.22405/2226-8383-2025-26-4-447-453
Abstract
This paper studies best simultaneous approximations of the second kind for linearly dependent irrationalities. It is shown that, starting from a certain point, they are determined by the convergents of the first irrationality, of which all others are multiples.
About the Authors
Mikhail Nikolaevich Dobrovol’skii
Geophysical centre of RAS
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University; Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, senior
researcher; junior researcher
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Bykovskii, V. A. 2002, “On the error of number-theoretic quadrature formulas”, Chebyshevskii Sbornik, 3(2(4)), pp. 27–33.
2. Voronoi, G. F. 1896, “On Generalization of the Algorithm of Continued Fraction”, Warsaw University.
3. Gorkusha, O. A. and Dobrovolsky, N. M. 2005, “On estimates of hyperbolic zeta function of lattices”, Chebyshevskii Sbornik, 6(2(14)), pp. 130–138.
4. Delone, B. N. 1947, St. Petersburg School of Number Theory, Moscow-Leningrad: Publishing House of the Academy of Sciences of the USSR, 422 p.
5. Delone, B. N. and Faddeev, D. K. 1940, “The theory of irrationalities of the third degree”, Trudy Matematicheskogo Instituta imeni V.A. Steklova, 11, pp. 3–340.
6. Dobrovolsky, M. N., Dobrovolsky, N. N. and Dobrovolsky, N. M. 2024, “Best joint Dirichlet approximations”, Chebyshevskii Sbornik, 25(5), pp. 1–15.
7. Kormacheva, A. N., Dobrovolsky, N. N., Rebrova, I. Y. and Dobrovolsky, N. M. 2021, “On the hyperbolic parameter of a two-dimensional lattice of comparisons”, Chebyshevskii Sbornik, 22(4), pp. 168–182.
8. Kormacheva, A. N., Dobrovolsky, N. N., Rebrova, I. Y. and Dobrovolsky, N. M. 2023, “On Bykovsky estimates for a measure of the quality of optimal coefficients”, Chebyshevskii Sbornik, 24(2), pp. 214–227.
9. Feldman, N. I. 1981, Approximation of Algebraic Numbers, Moscow: Moscow University Press, 200 p.
10. Khinchin, A. Ya. 1961, Continued Fractions, Moscow: Fizmatlit, 112 p.
Review
For citations:
Dobrovol’skii M.N., Dobrovol’skii N.N., Dobrovol’skii N.M. On a case of joint Dirichlet approximations. Chebyshevskii Sbornik. 2025;26(4):447-453. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-447-453
Views:
3
Email this article 