Discrete ergodic method of academician Yu. V. Linnik
https://doi.org/10.22405/2226-8383-2025-26-3-385-405
Abstract
This article is dedicated to the 110th anniversary of the birth of the outstanding world — famous mathematician Academician Yu. V. Linnik and his discrete ergodic method. First, biographical information about Yu. V. Linnik is given. Then, after a brief presentation of the necessary information from the arithmetic of quaternions including the theory of quaternion rotations constructed by B. A. Venkov, the very idea of the discrete ergodic method (hereinafter DEM), belonging to Yu. V. Linnik, is considered.
The next part of the article is devoted to the presentation of the ergodic theorem in the case of quaternions and its application to the question of the asymptotics of integer points over regions on a sphere with increasing radius.
After this applications of the DEM to indefinite ternary quadratic forms corresponding to cases of integer point distributions on hyperboloids, using second-order matrix arithmetic instead of quaternions.
The article concludes with a statement of some unsolved problems related of DEM and a list of references.
About the Author
Urusbi Mukhamedovich Pachev
Berbekov Kabardino-Balkarian State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Linnik, Y.V. 1938, “Generalization of Frobenius’ theorem and establishing its connection with Hurwitz’ theorem on the composition of quadratic forms”, Izv. AN SSSR. Ser. mat., Vol. 2, No. 1, pp. 41-52.
2. Linnik, Y.V. 1939, “Several new theorems on the representation of large numbers by individual positive ternary quadratic forms”, DAN SSSR, Vol. 24, No. 3, pp. 211-212.
3. Linnik, Y.V. 1939, “On the representation of large numbers by positive ternary quadratic forms”, DAN SSSR, Vol. 25, No. 7, p. 578.
4. Linnik, Y.V. 1939, “A general theorem on the representation of numbers by individual ternary quadratic forms”, Izv. AN SSSR. Ser. mat., Vol. 3, No. 1, pp. 87-108.
5. Linnik, Y.V. 1939, “On certain results relating to positive ternary quadratic forms”, Mat. sb., Vol. 5, Issue 3, pp. 453-471.
6. Linnik, Y.V. 1940, “On the representation of large numbers by positive ternary quadratic forms. Theses for the degree of Candidate of Physical and Mathematical Sciences”, Leningrad, 21 p.
7. Linnik, Y.V. 1940, “On the representation of large numbers by positive ternary quadratic forms”, Izv. AN SSSR, ser. matem., pp. 363-402.
8. Linnik, Y.V. 1967, Ergodic properties of algebraic fields, Leningrad, 208 p.
9. Venkov, B.A. 1922, “On the arithmetic of quaternions”, Publishing House of the Russian Academy of Sciences (Second report).
10. Malyshev, A.V. 1962, “On the representation of integers by positive quadratic forms”, Trudy Mat. in-ta im. V.A. Steklova AN SSSR, Vol. 65, Moscow-Leningrad, Publishing House of the Academy of Sciences of the USSR, 212 p.
11. Malyshev, A.V. 1975, “Yu. V. Linnik’s ergodic method in number theory”, Acta Arithmetica, Vol. XXII, pp. 555-598.
12. Cassels, J. 1982, Rational quadratic forms, Mir Publishers, Moscow, 440 p.
13. Wong, R. 1985, The Hardy-Littlewood method, Mir, Moscow.
14. Linnik, Y.V. 1949, “Quaternions and Cayley numbers: some applications of quaternion arithmetic”, Uspekhi mat. nauk, Vol. 4, Issue 5, pp. 49-98.
15. Linnik, Y.V. and Malyshev, A.V. 1953, “On integer points on the sphere”, DAN SSSR, No. 2, pp. 209-211.
16. Linnik, Y.V. 1954, “Asymptotic distribution of integer points on the sphere”, DAN SSSR, Vol. 96, No. 5, pp. 909-912.
17. Linnik, Y.V. 1955, “Asymptotic distribution of reduced binary quadratic forms in connection with Lobachevsky geometry”, Vestn. LGU, No. 2, Ser. mat., fiz., him., Issue 1, pp. 3-23.
18. Linnik, Y.V. 1956, “Asymptotic geometry of Gaussian genera; analogue of the ergodic theorem”, DAN SSSR, Vol. 108, No. 6, pp. 1018-1021.
19. Skubenko, B.F. 1962, “Asymptotic distribution of integer points on a one-sheeted hyperboloid and ergodic theorems”, Izv. AN SSSR. ser. matem., Vol. 26, No. 5, pp. 721-752.
20. Malyshev, A.V. and Pachev, U.M. 1980, “On the arithmetic of second-order matrices”, Zapiski nauchnykh seminarov LOMI, Vol. 93, pp. 87-141.
21. Malyshev, A.V. 1980, “On the application of the discrete ergodic method in the analytical arithmetic of indefinite ternary quadratic forms”, Zap. nauchn. semin. LOMI, Vol. 93, pp. 5-23.
22. Pachev, U.M. 1980, “On the distribution of integer points on some two-sheeted hyperboloids”, Zap. nauchn. semin. LOMI, Vol. 93, pp. 87-141.
23. Malyshev, A.V. and Nguyen Ngoc Goy 1983, “On the distribution of integer points on some one-sheeted hyperboloids”, Zap. nauchn. semin. LOMI, Vol. 121, pp. 83-93.
24. Karpov, A.N. 1986, “On the representation of integers by isotropic quadratic forms”, Zap. nauchn. semin. LOMI, Vol. 151, pp. 66-67.
25. Malyshev, A.V. and Shirokov, B.M. 1991, “A new proof of the key lemma of the discrete ergodic method for second-order vector-matrices”, Vestnik Leningr. un-ta, pp. 34-40.
26. Malyshev, A.V. 1981, “Discrete ergodic method and applications to the arithmetic of ternary quadratic forms”, Topics in classical number theory, Budapest, Vol. 34, pp. 1023-1049.
27. Pachev, U.M. 2006, “Representation of integers by isotropic ternary quadratic forms”, Izv. RAN. Ser. matem., Vol. 70, No. 3, pp. 167-184.
28. Linnik, Y.V. 1956, “Asymptotic geometry of Gaussian genera; an analogue of the ergodic theorem”, Dokl. AN SSSR, Vol. 108, No. 6, pp. 1018-1021.
29. Malyshev, A.V. and Pachev, U.M. 1979, “On the number of classes of integer positive binary quadratic forms whose arithmetic minimum is divisible by a given number”, Algebra i teoriya chisel, Nalchik, Issue 4, pp. 33-37.
30. Pachev, U.M. 1994, “On the number of classes of Gaussian genus whose arithmetic minimum is divisible by the square of a given odd number”, Matematicheskie zametki, Vol. 55, No. 2, pp. 118-127.
31. Pachev, U.M. 1997, “Ergodic properties of flows of positive binary quadratic forms in Gaussian genera”, Zap. nauchn. semin. LOMI RAN, Vol. 236, pp. 149-161.
32. Pachev, U.M. 2012, “On the distribution of reduced positive binary quadratic forms with the condition of divisibility of the first coefficients by residue classes”, Uchenye zapiski Orlovskogo universiteta, No. 6 (50), pp. 177-182.
33. Pachev, U.M. 2013, “On the distribution of reduced indefinite binary quadratic forms with the condition of divisibility of the first coefficients by residue classes”, Chebyshevskii sbornik, Vol. 14, Issue 2 (46), pp. 139-150.
34. Pachev, U.M. 2005, “Asymptotic distribution of classes of positive binary quadratic forms with conditions of divisibility of coefficients”, Fundamentalnaya prikladnaya matematika, Vol. 11 (6), pp. 123-130.
35. Pachev, U.M. 2007, “On the asymptotics of the number of reduced integer binary quadratic forms with the condition of divisibility of the first coefficients”, Sibirskii matematicheskii zhurnal, Vol. 48, No. 2, pp. 376-388.
36. Golubeva, E.P. 1970, “On the representation of large numbers by ternary quadratic forms”, Dokl. AN SSSR, Vol. 91, No. 3, pp. 519-521.
37. Bykovsky, V.A. 1985, “Arithmetico-analytic properties of binary positive definite quadratic forms”, Zap. nauchn. semin. LOMI, Vol. 144, pp. 5-20.
38. Duke, W. 1988, “Hyperbolic distribution problems and half-integral weight Maas forms”, Invent. math., Vol. 92, No. 1, pp. 78-90.
39. Golubeva, E.P. 1988, “Geodesics on the upper half-plane and distribution of quadratic irrationals”, Zap. nauchn. semin. LOMI RAN, Vol. 254, pp. 28-55.
40. Eichler, M. 1952, Quadratische Formen und orthogonale Gruppen, Berlin.
41. Peters, M. 1977, “Representations by definite ternary quadratic forms”, Acta arithm., Vol. 34, No. 1.
42. Teterin, Yu.G. 1983, “On the representation of integers by positive ternary quadratic forms”, Zap. nauchn. sem. LOMI, Vol. 121, Nauka Publishing House, Leningrad, pp. 117-156.
43. Teterin, Yu.G. 1985, “Asymptotic formula for the number of representations by completely positive ternary quadratic forms”, Izv. AN SSSR. Ser. matem., Vol. 49, Issue 2, pp. 393-426.
Review
For citations:
Pachev U.M. Discrete ergodic method of academician Yu. V. Linnik. Chebyshevskii Sbornik. 2025;26(3):385-405. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-385-405
Views:
23
Email this article 