§ 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.