# 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) |

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

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

[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 |

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

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

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Linnik discrete ergodic method.

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