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Ergodic Properties of Algebraic Fields (Paperback, Softcover reprint of the original 1st ed. 1968)
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Ergodic Properties of Algebraic Fields (Paperback, Softcover reprint of the original 1st ed. 1968)
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, 45
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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.
General
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