Ergodic properties of algebraic fields

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 x^2+y^2+z^2=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. The theory presented here arose from the works of Linnik [1] in 1940 on positive ternary quadratic forms (at that time the basic ergodic character of the phenomena dealt with was not yet realized). Many significant results in this region have been obtained by A. V. Malishev and B. F. Skubenko [2]. The reader is assumed to possess elementary knowledge of measure theory, probability theory, algebraic number theory, and the theory of matrices.