Discrepancy of general symplectic lattices

We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic lattice, the discrepancy $D(\Lambda,B_t)$ of the lattice point count with respect to the volumes is $O(vol(B_t)^{-\delta})$. This extends work of W. Schmidt who gave similar discrepancy bounds for the space of all lattices in $\mathbb{R}^n$.