On Values of Linear and Quadratic Forms at Integral Points

On an oppenheim-type conjecture for systems of quadratic forms

“…contains a number which is not badly approximable (we consider ∞ to be not badly approximable). In the case of one quadratic form, this was observed by Dani [Da00]. Theorem 1 below is an analog of this fact in dimension d > 2.…”

Oppenheim conjecture for pairs consisting of a linear form and a quadratic form

“…For d = 3, condition (1) implies that the surfaces {Q = 0} and {L = 0} intersect transversally. Therefore, it follows from the result of Dani [Da00] mentioned above that the analogue of Theorem 1 does not hold for d = 3.…”

“…One can hope to remove the second condition. However, Dani proved in [Da00] that if the surface {Q = 0} and the plane {L = 0} intersect transversally, the density can fail for a set pairs of full Hausdorff dimension. On the other hand, it is easy to see using Moore ergodicity criterion that the density holds for a set of pairs of full measure provided that the surfaces {Q = 0} and {L = 0} have nonzero intersection.…”

A generic effective Oppenheim theorem for systems of forms

“…This was generalised by Gorodnik [16] to pairs (Q, L) in four or more variables. Further work on systems of forms has been done by Gorodnik [17] for systems of quadratic forms, by Dani [5,6] for systems comprising a quadratic form and a linear form, and by Müller [22,23] for certain systems of quadratic forms. Other than the papers of Müller which use the circle method and therefore get quantitative results, the other works use homogeneous dynamics and establish qualitative statements.…”