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(Zd) 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∈Zd}⊂R2 is dense in R2 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 Qi, i=1,…,s, be nondegenerate indefinite quadratic forms of dimension d. Under some conditions on the intersection of zero surfaces {Qi=0}, i=1,…,s, we compute measure and Hausdorff dimension of the set of g∈ GL(d,R) such that the set {(Q1(gx),…,Qs(gx)):x∈Zd}⊂Rs 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, Γ a lattice in G, and H a noncompact closed subgroup of G. We consider the action of Γ on G/H. Fix a norm ||·|| on G. For T>0, Ω⊆G/H, and x0∈G/H, define N T (Ω,x 0)=#{γ∈Γ:γ·x 0 ∈Ω,||γ||<T}. Distribution of Γ-orbits in G/H is reflected in the asymptotic behavior of the quantity NT (Ω,x0) as T→∞. We determine this asymptotic behavior in several cases:
A lattice in SL (d,R) acting on the Furstenberg boundary of SL (d,R);
A lattice in SL (d,R) acting on the space of k-frames in Rd (1≤k≤d-1);
A lattice in a general matrix Lie group G acting on a factor space G/H where H is either semisimple or unipotent subgroup of G that satisfies some technical conditions.
Also we deduce some interesting corollaries concerning distribution of values of the Gram matrix of a quadratic form on integer frames and uniform distribution of orbits of SL(d,Z) in the torus Td.