On the number of points of an incomplete lattice in rectangular regions
https://doi.org/10.22405/2226-8383-2024-25-4-138-146
Abstract
In 2013–2015 it was shown that for any purely real algebraic irrationality 𝛼, starting from some place, all residual fractions in the expansion of 𝛼 into a continued fraction will appear to be the reduced algebraic irrationalities.
We construct the examples of purely real algebraic irrationalities 𝛼 for which this number of the residual fraction is arbitrarily large.
Supporting agencies: The study was carried out within the framework of state task No. 073-03-2022-117/7 on the topic “Numbertheoretic methods in approximate analysis and their applications in mechanics and physics” .
About the Authors
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Mikhailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Dobrovol’skii, N.M. & Dobrovol’skii, N.N. 2015, “About minimal polynomial residual fractions for algebraic irrationalities”, Chebyshevskii Sbornik, vol. 16, no. 3(55), pp. 147–182. (Russian)
2. Dobrovol’skii, N.M., Dobrovol’skii, N.N., Sobolev, D.K. & Soboleva, V.N. 2017, “Classification of pure-real algebraic irrationalities”, Chebyshevskij sbornik, vol. 18, no. 2, pp. 98–128.
3. Dobrovol’skii, N.M., Sobolev, D.K. & Soboleva, V.N. 2013, “On a matrix decomposition of the reduced cubic irrational”, Chebyshevskii sbornik, vol. 14, no. 1, pp. 34–55.
4. Matiyasevich, Yu.V. 1974, “Existence of non-effectivizable estimates in the theory of exponentially Diophantine equations”, Zap. scientific. sem. LOMI, 40, pp. 77–93.
5. Matiyasevich, Yu.V., 1993, “Hilbert’s tenth problem”, M.: Science, Fizmatlit.
6. Matiyasevich, Yu.V., 2011, “What can and cannot be done with Diophantine problems”, Proc. Steklov Inst. Math., 275, pp. 118–132.
7. Trikolich, E. V. & Yushina, E. I. 2009, “Continued fractions for quadratic irrationalities from the field Q(√5)”, Chebyshevskii Sbornik., vol. 10, no. 1(29), pp. 77–94. (Russian)
8. Khinchin A. Y., 1949, “Continued fractions”, M.; L.: Gostekhizdat.
Review
For citations:
Dobrovol’skii N.N., Dobrovol’skii N.M. On the number of points of an incomplete lattice in rectangular regions. Chebyshevskii Sbornik. 2024;25(4):138-146. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-4-138-146
Views:
59
Email this article 