Buterus, Paul: Value Distribution of Quadratic Forms and Diophantine Inequalities. 2020
Inhalt
- 1 Introduction
- 1.1 The Oppenheim Conjecture: A Short Historical Overview
- 1.2 Integer-valued Quadratic Forms
- 1.3 Main Result on Diagonal Indefinite Forms
- 1.4 Our Contribution to the Non-Diagonal Case
- 1.4.1 Value Distribution of Quadratic Forms
- 1.4.2 Quantitative Bounds for Diophantine Inequalities
- 1.4.3 Diophantine Quadratic Forms
- 1.5 Recent Development on Generic Variants of the Oppenheim Conjecture
- 1.6 Further Research Questions and Open Problems
- 1.7 Notation and Glossary
- 2 Indefinite Diagonal Quadratic Forms
- 2.1 Sketch of Proof
- 2.2 Fourier Analysis
- 2.3 First Coupling via Diophantine Approximation
- 2.4 Iteration of the Coupling Argument
- 2.5 Proof of Theorem 1.6: Counting Approximants
- 3 Distribution of Values of Quadratic Forms
- 3.1 Effective Estimates
- 3.2 Organization and Sketch of Proof
- 3.2.1 Smooth Weights on Zd
- 3.2.2 First Steps of the Proof
- 3.2.3 Mean-Value Estimates
- 3.2.4 The Role of the Region Omega
- 3.3 Lattice Point Remainder via Fourier Representation
- 3.3.1 Smooth Approximation of the Indicator Function of [a,b]
- 3.3.2 Rewriting of the Remainder Term
- 3.3.3 Splitting the Fourier Integrals
- 3.4 Special Symplectic Lattices
- 3.4.1 Structure of Symplectic Lattices
- 3.4.2 Approximation by Compact Subgroups
- 3.4.3 Application to the Lattice Remainder
- 3.5 Smoothing of Special Parallelepiped Regions
- 3.5.1 Fourier Transform of Weights for Polyhedra
- 3.5.2 Lattice Point Remainders for Admissible Parallelepipeds
- 3.6 Proof of Theorem 3.1
- 3.7 Applications of Theorem 3.1
- 4 General Indefinite Quadratic Forms
- 4.1 Quadratic Forms of Diophantine Type (k,A)
- 4.2 Irrational and Diophantine Lattices
- 4.3 Proofs of Theorems 1.9, 1.11 and Corollaries 4.3, 4.4
- 4.4 Davenport-Lewis Conjecture
- 5 Appendix A
- 5.1 Mean-Value Estimates for Quadratic Exponential Sums
- 5.2 A Refined Variant of Weyl's Inequality
- 5.3 Smoothing Kernels
- 6 Appendix B
- 6.1 Fourier Analysis, Smoothing and Theta-Series
- 6.1.1 Estimates for the Theta-Series
- 6.1.2 Estimation of I(vartheta) and I(Delta)
- 6.1.3 Rewriting of I(theta)
- 6.2 Margulis' Averaging Result
- 7 Appendix C
- 7.1 Integer-valued Quadratic Forms
- 7.2 Schlickewei's Work on Small Zeros of Integral Quadratic Forms
- 7.3 Discrete Optimization: Possible Signatures and Exponents
- References