  Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient
conditions under which certain translates of homogeneous measures converge, and
they determined the limiting measures in the cases of convergence. The class of
measures they considered includes the maximal horospherical measures. In this
paper we prove the corresponding effective equidistribution results in the
space of unimodular lattices. We also prove the corresponding results for
probability measures with absolutely continuous densities in rank two and
three. Then we address the problem of determining the error terms in two
counting problems also considered by Mohammadi and Salehi-Golsefidy. In the
first problem, we determine an error term for counting the number of lifts of a
closed horosphere from an irreducible, finite-volume quotient of the space of
positive definite $n\times n$ matrices of determinant one that intersect a ball
with large radius. In the second problem, we determine a logarithmic error term
for the Manin conjecture of a flag variety over $\mathbb{Q}$.
