Dynamics on homogeneous spaces and applications to simultaneous diophantine approximation
We give some new results about diophantine simultaneous inequalities involving one quadratic form and one linear generalising the Oppenheim conjecture. In the first part we compute an exact lower asymptotic estimate of the number of integral values taken by such pairs, by using uniform distribution of unipotents flows. In the second part, we prove an S-adic version of the Oppenheim type problem for pairs. The proof uses S-adic dynamics and strong approximation. We also discuss a conjecture due to A. Gorodnik about�finding optimal conditions which ensure density for pairs in dimension greater than three. This conjecture was partially the motivation of this thesis and is still open at this time.
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