On lattices of simultaneous Dirichlet approximations
https://doi.org/10.22405/2226-8383-2026-27-1-63-76
Abstract
This paper studies the properties of unimodular lattices of simultaneous Dirichlet approximations and reciprocal lattices of simultaneous Dirichlet approximations. A theorem is proved stating the equality of distances between two lattices and between two corresponding reciprocal lattices. The completeness of the spaces of lattices of simultaneous Dirichlet
approximations and mutual lattices of simultaneous Dirichlet approximations is proved.
About the Authors
Alexander Petrovich Krylov
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
postgraduate student
Nikolai Mikhailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Irina Nikolaevna Balaba
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Dobrovol’skaya, L. P., Dobrovol’skii, M. N., Dobrovol’skii, N. M. & Dobrovol’skii, N. N. 2012, “The hyperbolic Zeta function of grids and lattices, and calculation of optimal coefficients”, Chebyshevskii sbornik, vol. 13, no. 4(44), pp. 4-–107.
2. Kassels, D. 1965, Vvedenie v geometriyu chisel, [Introduction to the geometry of numbers], Mir, Moscow, (Russia).
3. Krylov, A. P., Dobrovolsky, N. M. 2022, “Metric space of two-dimensional diagonal unimodular lattices”, Notes of scientific seminars of the Tula School of Number Theory, Iss. 1, pp. 37–41.
4. A. P. Krylov, 2025-2026, "The Metric Space of Diagonal Unimodular Lattices" , Notes of Scientific Seminars of the Tula School of Number Theory, Iss. 3, pp.
5. E. N. Smirnova, O. A. Pikhtilkova, N. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2023, “Smooth variety of lattices”, Chebyshevskii sbornik, vol. 24, no. 4, pp. 299–310.
6. Dobrovol’skaya, L. P., Dobrovol’skii, M. N., Dobrovol’skii, N. M. & Dobrovol’skii, N. N. 2014, “On Hyperbolic Zeta Function of Lattices”, Continuous and Distributed Systems. Solid Mechanics and Its Applications, vol. 211, pp. 23–62.
Review
For citations:
Krylov A.P., Dobrovol’skii N.M., Balaba I.N. On lattices of simultaneous Dirichlet approximations. Chebyshevskii Sbornik. 2026;27(1):63-76. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-1-63-76
Views:
350
JATS XML
Email this article 