# Linnik discrete ergodic method

A special method in analytic number theory that uses non-commutative arithmetic and reduces questions on the uniform distribution of integral points on a manifold to the consideration of "flows" on integral points on this manifold and the operators that create these "flows" . The foundations of the method were laid by Yu.V. Linnik [1]. Linnik's discrete ergodic method derives its essential and "ergodic" features from the character of its results [2], [3]. The discrete ergodic method has been applied to questions on the asymptotic distribution of integral points over the surface of the corresponding ellipsoid or hyperboloid. The best known result is Linnik's theorem on the asymptotic uniform distribution of integral points over the surfaces of spheres of increasing radius (see [2], Chapt. IV).

#### References

[1] | Yu.V. Linnik, "On the representation of large numbers by positive ternary quadratic forms" Izv. Akad. Nauk SSSR Ser. Mat. , 4 : 4–5 (1940) pp. 363–402 (In Russian) MR0002347 Zbl 0022.20105 Zbl 0022.00706 Zbl 65.1148.01 Zbl 65.1147.01 |

[2] | Yu.V. Linnik, "Ergodic properties of algebraic fields" , Springer (1968) (Translated from Russian) MR0238801 Zbl 0162.06801 |

[3] | A.V. Malyshev, "The representation of integers by positive quadratic forms" Trudy Mat. Inst. Steklov. , 65 (1962) Zbl 0481.10018 Zbl 0425.10024 Zbl 0163.04604 Zbl 0096.02903 |

[4] | A.V. Malyshev, "A new version of the ergodic method of Yu.V. Linnik in number theory" J. Soviet Math. , 11 (1978) pp. 346–352 Zap. Nauchn. Sem. Leningr. Otdel. Mat. Inst. , 50 (1975) pp. 179–186 Zbl 0403.10021 |

[5] | A.V. Malyshev, "Yu.V. Linnik's ergodic method in number theory" Acta Arithm. , 27 (1975) pp. 555–598 MR0421950 MR0421942 MR0371815 Zbl 0303.10020 Zbl 0301.01011 Zbl 0299.01016 |

[6] | M. Peters, "Darstellungen durch definite ternäre quadratische Formen" Acta Arithm. , 34 (1977) pp. 57–80 MR0476632 Zbl 0323.10017 |

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Linnik discrete ergodic method.

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