OPPENHEIM CONJECTURE

The following sections are included:

• Analytic Number Theory Methods

  • Chowla's result and the distribution of values of positive definite quadratic forms at integer points

  • Diagonal forms in n≥5 variables

  • A modification of the theorem of Davenport and Heilbronn

  • Results for general quadratic forms

• Oppenheim Conjecture And Subgroup Actions On Homogeneous Spaces

  • Paper [CaS] of Cassels and Swinnerton-Dyer

  • Paper [CaS] (continuation)

  • Closures of orbits of orthogonal groups and integer solutions of quadratic inequalities

  • Quantitative results

  • Markov spectrum

• Unipotent Flows On Homogeneous Spaces

  • Growth properties of polynomials

  • Unipotent groups of linear transformations

  • Recurrence to compact sets

  • Finiteness of ergodic measures

  • General properties of minimal invariant sets

  • Topological limits of double cosets in algebraic groups

  • Proof of Theorem 2.3.4 (a)

  • Conjectures of Dani and Raghunathan. Uniform distribution

  • Ratner's results on unipotent flows

• References