On an oppenheim-type conjecture for systems of quadratic forms

Abstract

Let Q i , i = 1, . . . , t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set. As a corollary, we deduce several results on the magnitude of the set ∆ of g ∈ GL(d, R) such that the closure of the set {(Q 1 (gx), . . . , Q t (gx)) :x ∈ Z d − {0}} contains (0, . . . , 0). Special cases are described when depending on the mutual position of the hypersurfaces {Q i = 0}, i = 1, . . . , t, the set ∆ has full Haar measure or measure zero and Hausdorff dimension d 2 − d−2 2 .

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.…”

On the density at integer points of a system comprising an inhomogeneous quadratic form and a linear form

“…The work of Dani and Margulis was generalised by Gorodnik [13] who studied pairs comprising a quadratic and linear form in dimensions greater than 3. Subsequently, he studied systems of quadratic forms in [14]. Further progress on systems comprising a quadratic and linear form was made in [6] by Dani.…”

Density of values of linear maps on quadratic surfaces

“…The case when P consists of a system of many linear forms and a quadratic form has been considered by S.G. Dani in [Dan08]. A. Gorodnik also considered the case when P consists of a system of quadratic forms in [Gor04a]. To the authors knowledge the case when X = R d has not been considered.…”