Equidistribution of expanding degenerate manifolds in the space of lattices

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real‐analytic submanifold under a diagonal flow. This work extends the earlier result of Shah in the case of nondegenerate submanifolds. We apply the above dynamical result to show that if the affine span of a real‐analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet‐improvable.