Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation

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The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and p -adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock–Margulis (strong extremality of nondegenerate submanifolds of ℝ^n ) are generalized to the S -arithmetic setting.