2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161184We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $\Bbb R^n$ and their nondegenerate submanifolds.26 pages, a revised version. Trans. AMS, to appearDynamical SystemsNumber Theory22F30; 11J83text