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.
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ISBN-13
978-3-540-04101-6 (9783540041016)
DOI
10.1007/978-3-642-86631-9
Schweitzer Klassifikation
§ 1 Ergodic Theory.- § 2 Applications of Ergodic Concepts to the Theory of Diophantine Equations.- I. A Summary of Elementary Ergodic Theory and Limit Theorems of Probability Theory.- § 1 Basic Theorems of Ergodic Theory.- § 2 Applications to Metric Number Theory.- § 3 Limit Theorems of Probability Theory.- II. A Summary of the Arithmetic of Quaternions and Matrices.- § 1 Arithmetic of Quaternions.- § 2 Arithmetic of 2 × 2 Matrices.- § 3 Arithmetic of n × n Matrices.- III. Rotations of the Sphere, Binary Quadratic Forms, and Quaternions.- § 1 Supplementary Arithmetic Information.- § 2 Asymptotic Properties of Rotations of a Large Sphere.- IV. Asymptotic Geometrical and Ergodic Properties of the Set of Integral Points on the Sphere.- § 1 Formulation of the Problem.- § 2 Ergodic Properties.- § 3 Primitive Points in the Fundamental Triangle.- § 4 Reduction of the Problem to the Calculation of Probabilities of Large Deviations.- § 5 Calculation of Probabilities of Large Deviations. An Application of Theorem III.2.1.- § 6 Completion of the Proof to the Ergodic Theorem IV.2.1.- § 7 Orthogonal Matrices. A Mixing Theorem. The Asymptotic Distribution of Primitive Points on the Sphere.- § 8 Supplementary Remarks.- V. Flows of Primitive Points on a Hyperboloid of Two Sheets. Asymptoticity of Reduced Binary Forms in Connection with Lobachevskian Geometry.- § 1 Formulation of the Problem.- § 2 Formulation of the Basic Theorems.- § 3 Formulation of the Basic Lemma.- § 4 Continuation of the Proof of the Basic Lemma.- § 5 Study of Rotations.- § 6 Behavior of Senior Forms.- § 7 An Estimate for the Number of Primitive Representations.- § 8 A Lemma on Divisibility of Matrices in Connection with Probabilities of Large Deviations.- § 9 Reduced Forms with Small First Coefficients.- §10 Transition of the Proof of Theorem V.2.1.- §11 A Lemma on Matrices.- § 12 A Lemma due to I. M. Vinogradov and Kloosterman Sums.- §13 Consequences of Lemma V. 11.1.- §14 Asymptotic Geometry of Hyberbolic Rotations.- §15 Evaluation of Probabilities.- §16 Proof of Theorem V.2.1.- §17 Proofs of Theorems V.2.2. and V.2.1.- § 18 On Ergodic Theorems for the Flow of Primitive Points of the Hyperboloid of Two Sheets.- §19 Ergodic Theorems for a Modular Invariant.- §20 Supplementary Remarks.- VI. Flows on Primitive Points on a Hyperboloid of One Sheet.- § 1 Formulation of the Problem.- § 2 Formulation of the Basic Theorem. A Lemma on Integral Points.- § 3 Asymptoticity of Hyperbolic Rotations.- § 4 Further Investigation of the Asymptoticity of Hyperbolic Rotations.- § 5 An Ergodic Theorem and a Mixing Theorem.- VII. Algebraic Fields of a More General Type.- § 1 General Remarks.- § 2 On the Representations of Algebraic Numbers by Integral Matrices.- § 3 Rotations.- VIII. Asymptotic Distribution of Integral 3 × 3 Matrices.- § 1 Formulation of the Problem.- § 2 Some Estimates.- § 3 Completion of the Proof.- IX. Further Generalizations. A Connection with the Generalized Riemann Hypothesis.- § 1 Further Generalizations.- § 2 A Connection with the Generalized Riemann Hypothesis and its Weaker Forms.- § 3 Elementary Ergodic Considerations.- X. An Arithmetic Simulation of Brownian Motion.- § 1 General Remarks. Formulation of the Problem.- § 2 Basic Theorems.- XI. Supplementary Remarks. Problems.- Author Index.
