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Number-theoretic method in approximate analysis. (Russian. English summary) Zbl 1434.11136

Summary: Into the image it is considered issues of history and the modern development of number-theoretic method in the approximate analysis which based in the work of N. M. Korobov and his disciples. It is reviewed the connection of the theory of uniform distribution and theoretical-numeric method in approximate analysis. It is shown that the condition for the theoretical-numeric method was the integral criterion G. Weyl. It is disassembled main types of number-theoretic nets: uneven, parallelepipedal and algebraic. It is consecrated the activities of the workshop three K, it is explored the biographical information about N. M. Korobov and brief information about the leaders of the seminar and its participants.
It is described the main directions of research in theoretical-numeric method in approximate analysis. It is examined the issues of information security theoretic-numeric method in approximate analysis using POIS TMK.
More detailed it is outlined the issues of finding the optimal coefficients for parallelepipedal nets, the theory of the hyperbolic Zeta function of lattices, the theory of algebraic nets and its relationship with the theory of Diophantine approximations.
In particular, we discuss the algebraic theory of polynomials Tue. The theory is based on the study of submodules of \(\mathbb{Z}[t]\)-module \(\mathbb{Z}[t]^2\). It is considered of submodules that are defined by one defining relation and one defining relation \(k\)-th order. More complex submodule is the submodule given by one polynomial relation. Sub par Tue \(j\)-the order are directly connected with polynomials Tue \(j\)-th order. Using the algebraic theory of pairs of submodules of Tue \(j\)-th order is managed to obtain a new proof of the theorem of M. N. Dobrowolski (senior) that for each \(j\) there are two fundamental polynomial Tue \(j\)-th order, which are expressed through others. Basic polynomials are determined with an accuracy of unimodular polynomial matrices over the ring of integer polynomials.
It is discussed the fractional-linear transformation of TDP-forms. It is shown that the transition from TDP-forms associated with an algebraic number \(\alpha\) to TDP-the form associated with the residual fraction to algebraic number \(\alpha \), TDP-form is converted under the law, similar to the transformation of minimal polynomials and the numerators and denominators of the respective pairs of Tue is converted using the linear-fractional transformations of the second kind. Besides, we discuss the new classification of purely real algebraic irrationalities which based on their expansion in continued fractions. It is shown that for purely real algebraic irrationalities \(\alpha\) of degree \(n\ge2\), starting from some numbers \(m_0=m_0(\alpha)\), the sequence of residual fractions \(\alpha_m\) is a sequence given the algebraic irrationalities.
It is found recurrence the formula for finding the minimal polynomials of the residual fractions using the linear-fractional transformations. The compositions of these linear-fractional transformations is a linear-fractional transformation that maps the system conjugate to algebraic irrationascenic spots \(\alpha\) in the system of associated to the residual fraction, with a pronounced effect of concentration nearly rational fraction \(-\frac{Q_{m-2}}{Q_{m-1}} \).
It is established that the sequence of minimal polynomials for residual fractions forms a sequence of polynomials with equal discriminants.
Lists some of the most pressing unsolved problems.

MSC:

11J82 Measures of irrationality and of transcendence
11K06 General theory of distribution modulo \(1\)
11K45 Pseudo-random numbers; Monte Carlo methods
11H06 Lattices and convex bodies (number-theoretic aspects)
11K50 Metric theory of continued fractions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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References:

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[103] Dobrovol’skii N. M., Rebrova I. Yu., Roshhenya A. L., “Continuity of the hyperbolic Zeta function of lattices”, Matematicheskie zametki, 63:4 (1998), 522-526 · Zbl 0924.11075 · doi:10.4213/mzm1312
[104] Dobrovol’skii N. M., Rodionova O. V., “Quadrature formulas with generalized parallelepipedal grids”, Modern problems of number theory and its applications, Proceedings of the third International conference (Tula, Russia), 1996, 47-48
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[106] Dobrovol’skii N. M., Rodionova O. V., “On one finite Fourier series and its applications”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 4:3 (1998), 68-79
[107] Dobrovol’skii N. M., Roshhenya A. L., “On the number of lattice points in a hyperbolic cross”, Algebraic, probabilistic, geometric, combinatorial and functional methods in number theory, Proceedings of the II international conference (Voronezh, Russia), 1995, 53
[108] Dobrovol’skii N. M., Roshhenya A. L., “On the analytic continuation of the hyperbolic Zeta function of rational lattices”, Modern problems of number theory and its applications, Proceedings of the III international conference (Tula, Russia), 1996, 49
[109] Dobrovol’skii N. M., Roshhenya A. L., “On continuity of the hyperbolic Zeta function of lattices”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 2:1 (1996), 77-87
[110] Dobrovol’skii N. M., Sobolev D. K., Soboleva V. N., “On a matrix decomposition of the reduced cubic irrational”, Chebyshevskij sbornik, 14:1 (2013), 34-55 · Zbl 1429.11128
[111] Dobrovol’skii N. N., “On the number of integer points in a hyperbolic cross at the values of \(1\le t<21\)”, Izvestiya TulGU. Seriya Matematika. Mekhanika. Informatika, 9:1 (2003), 91-95
[112] Dobrovol’skii N. N., “A trigonometric polynomial on a grid of Smolyak”, Proceedings of the international scientific conference «Modern problems of mathematics, mechanics, computer science» (Tula, Russia), 2007, 34-36
[113] Dobrovol’skii N. N., “Deviation of two-dimensional Smolyak grids”, Chebyshevskij sbornik, 8:1(21) (2007), 110-152 · Zbl 1341.11045
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