Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge

Ergodic Properties of Algebraic Fields

Authors: Linnik, Yurij V.

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About this Textbook

The applications of ergodic theory to metric number theory are well known; part of the latter theory turns out to be essentially a special case of general ergodic theorems. In the present book other applications of ergodic concepts are presented. Constructing "flows" of integral points on certain algebraic manifolds given by systems of integral polynomials, we are able to prove individual ergodic theorems and mixing theorems in certain cases. These theorems permit asymptotic calculations of the distributions of integral points on such manifolds, and we arrive at results inaccessible up to now by the usual methods of analytic number theory. Typical in this respect is the theorem concerning the asymptotic distribution and ergodic behavior of the set of integral points on the sphere X2+ y2+z2=m for increasing m. It is not known up until now how to obtain the simple and geometrically obvious regularity of the distribution of integral points on the sphere other than by ergodic methods. Systems of diophantine equations are studied with our method, and flows of integral points introduced for this purpose turn out to be closely connected with the behavior of ideal classes of the corresponding algebraic fields, and this behavior shows certain ergodic regularity in sequences of algebraic fields. However, in this book we examine in this respect only quadratic fields in sufficient detail, studying fields of higher degrees only in chapter VII.

Buy this book

Softcover $99.00
price for USA
  • ISBN 978-3-642-86633-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.

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Bibliographic Information

Bibliographic Information
Book Title
Ergodic Properties of Algebraic Fields
Authors
Translated by
Keane, M.S.
Series Title
Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge
Series Volume
45
Copyright
1968
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
Softcover ISBN
978-3-642-86633-3
Edition Number
1
Number of Pages
X, 194
Additional Information
Title of the original Russian edition: Ergodiceskie svojstva algebraiceskich poleij
Topics