On 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 {(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we deduce several results on the magnitude of the set \Delta of g\in GL(d,R) such that the closure of the set {(Q_1(gx),...,Q_t(gx)): x\in 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 \Delta has full Haar measure or measure zero and Hausdorff dimension d^2-(d-2)/2.