In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular we show that this set is dense in the range of the linear map subject to certain algebraic conditions on the linear map and the quadratic form that defines the surface. The proof uses Ratner's Theorem on orbit closures of unipotent subgroups acting on homogeneous spaces.

Abstract. In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular, it is shown that subject to certain algebraic conditions, this set is equidistributed. This can be thought of as a quantitative version of the main result from [Sar11]. The methods used are based on those developed by A. Eskin, S. Mozes and G. Margulis in [EMM98]. Specifically, they rely on equidistribution properties of unipotent flows.

We study the dynamics of SL3(R) and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of R 3 . We focus on the case where the acting group is Zariski dense in either SL3(R) or SO(2, 1)(R). Using techniques of Benoist and Quint we prove that for a compactly supported probability measure µ on SL3(R) whose support generates a group which is Zariski dense in SL3(R), there exists a unique µ-stationary probability measure on X. When the Zariski closure is SO(2, 1)(R) we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.proximally on R 3 , then P µ (Gr 2 (R 3 )) consists of a single element. We will refer to it as the Furstenberg measure of µ on Gr 2 (R 3 ) and denote it byν Gr 2 (R 3 ) .Remark 1.4. It will be important for us that the Furstenberg measure is atom free. This is ensured by the strong irreducibility assumption, since if there was an atom ofν Gr 2 (R 3 ) then the set of atoms with maximal weight is a finite Γ µ -invariant set.We fix {e 1 , e 2 , e 3 } the standard orthonormal basis of unit vectors in R 3 . For v ∈ R 3 and 1 ≤ i ≤ 3 we will write v i := v, e i . As before we consider the indefinite quadratic form Q :(1.1) Let H µ denote the Zariski closure of Γ µ . In what follows we will concentrate on two cases which will be referred to as Case I and Case II as follows:

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Abstract. In this paper we investigate the distribution of the set of values of a quadratic form Q, at integral points. In particular we are interested in the n-point correlations of the this set. The asymptotic behaviour of the counting function that counts the number of n-tuples of integral points (v 1 , . . . , vn), with bounded norm, such that the n − 1 differences, lie in prescribed intervals is obtained. The results are valid provided that the quadratic form has rank at least 5, is not a multiple of a rational form and n is at most the rank of the quadratic form. For certain quadratic forms satisfying Diophantine conditions we obtain a rate for the limit. The proofs are based on those in the recent preprint ([GM13]) of F. Götze and G. Margulis, in which they prove an 'effective' version of the Oppenheim Conjecture. In particular, the proofs rely on Fourier analysis and estimates for certain theta series.