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 a previous paper. The methods used are based on those developed by A. Eskin, S. Mozes and G. Margulis. Specifically, they rely on equidistribution properties of unipotent flows.
"Equidistribution of values of linear forms on quadratic surfaces." Algebra Number Theory 8 (4) 895 - 932, 2014. https://doi.org/10.2140/ant.2014.8.895