Table of contents

CONTENTS §1. Introduction §2. Statement of the result Chapter I. The simplest properties of geodesic flows and Laplace-Beltrami operators on Liouville surfaces §3. The geodesic flow on a Liouville surface §4. The Laplace-Beltrami operator on a Liouville surface and its eigenfunctions Chapter II. Asymptotic analysis of eigenfunctions and eigenvalues §5. Integrable quantum systems: introduction §6. Asymptotic formulae for eigenvalues. Geometric interpretation of the spectral function §7. Asymptotic analysis of the system (4.3): reduction to a standard form §8. Asymptotic representations of the solutions of equation (7.5) in Weber functions §9. Proof of Theorem 6.1: derivation of the quantization rules Chapter III. Asymptotic behaviour of the number of points of a two-dimensional lattice in a family of homothetic domains §10. Survey of number-theoretic results §11. Statistical properties of the function : statement of the result and idea of the proof §12. The Poisson summation formula and other auxiliary results §13. Application of the method of stationary phase §14. Existence of a limit distribution for the function §15. Analytic properties of the limit distribution §16. Proof of Theorems 11.3 and 11.4 §17. Proof of the auxiliary assertions §18. Proof of Theorems 2.1 and 6.2 Appendix I. Weber functions (parabolic cylinder functions) Appendix II. Besicovitch almost periodic functions Appendix III. Polars of curves and their simplest properties References

CONTENTS Introduction. Formulation of the fundamental theorem Chapter I. Weyl sums and the estimate of the remainder term in the law of distribution of the fractional parts of a polynomial §1. Fundamental definitions and notation §2. Auxiliary lemmas and theorems §3. Formulation and proof of Theorem 3.1 §4. Estimates of Weyl sums §5. Estimates of the remainder term in the law of distribution of the fractional parts of a polynomial Chapter II. Joint distribution of the fractional parts of a polynomial §1. Fundamental definitions §2. Auxiliary lemmas §3. Uniform joint distribution of the fractional parts of a polynomial and the asymptotic law of distribution of the distance between neighbouring fractional parts of a polynomial §4. Auxiliary theorem §5. Estimates of the remainder term in the law of joint distribution of the fractional parts of a polynomial References