Ergodic Properties of Algebraic Fields

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 SmallFirst 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.