An extension of quantitative nondivergence and applications to Diophantine exponents

We 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 ${\mathbb{R}}^n$ and their nondegenerate submanifolds.