# Density and equidistribution of integer points

Abstract (Summary)

Celebrated work of G. Margulis on the conjecture of Oppenheim has established that for a real nondegenerate indefinite quadratic form Q in dimension d?3 , which is not a scalar multiple of a rational form, the set Q(Z d ) is dense in R . We extend this result to the case of a pair consisting of a quadratic form Q and a linear form L . Namely, we show that the set {(Q(x),L(x)):x?Z d }?R ^2 is dense in R ^2 provided that some natural algebraic conditions are satisfied. We also consider a similar, but much more complicated problem concerning values of a system of quadratic forms at integer points. Let Q i , i=1,…,s , be nondegenerate indefinite quadratic forms of dimension d . Under some conditions on the intersection of zero surfaces {Q i =0} , i=1,…,s , we compute measure and Hausdorff dimension of the set of g? GL (d,R) such that the set {(Q 1 (gx),…,Q s (gx)):x?Z d }?R s accumulates at (0,…,0) . In some cases, this Hausdorff dimension is fractional. Using recent fundamental results on rigidity of unipotent flows, we investigate distribution of orbits of a lattice (of a Lie group) in a homogeneous space. More precisely, let G be a matrix Lie group, ? a lattice in G , and H a noncompact closed subgroup of G . We consider the action of ? on G/H . Fix a norm ||·|| on G . For T>0 , ??G/H , and x 0 ?G/H , define N T (?,x 0 )=#{???:?·x 0 ??,||?||