On an oppenheim-type conjecture for systems of quadratic forms

Abstract

LetQ i, i=1, …,t, be real nondegenerate indefinite quadratic forms ind variables. We investigate under what conditions the closure of the set $$\left\{ {\left( {Q_1 \left( {\bar x} \right),...,Q_t \left( {\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). As a corollary, we deduce several results on the magnitude of the set Δ ofg ∈ GL(d, ℝ) such that the closure of the set $$\left\{ {\left( {Q_1 \left( {g\bar x} \right),...,Q_t \left( {g\bar x} \right)} \right):\bar x \in \mathbb{Z}^d - \left\{ {\bar 0} \right\}} \right\}$$ contains (0, …, 0). Special cases are described when, depending on the mutual position of the hypersurfaces {Q i=0},i=1, …,t, the set Δ has full Haar measure or measure zero and Hausdorff dimensiond 2−(d−2)/2.