I have been working on a problem related to the the so called 'Oppenheim Conjecture'. The first article below investigates 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.
In the second article below, 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 the first article. The methods used are based on those developed by A. Eskin, S. Mozes and G. Margulis in [MR1609447]. Specifically, they rely on equidistribution properties of unipotent flows.