  The present paper analyzes the discrepancy of distribution of rational points
on general semisimple algebraic group varieties. The results include
mean-square, almost sure, and uniform discrepancy estimates with explicit error
bounds, which apply to general families of subsets, and are valid at
arbitrarily small scales. We also consider an analogue of W. Schmidt's
classical theorem, which establishes effective almost sure asymptotic counting
of rational solutions to Diophantine inequalities in Euclidean spaces. We
formulate and prove a version of it for rational points on the group variety,
together with an effective bound which in some instances can be expected to be
best possible.
