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 ℝn) are generalized to the S-arithmetic setting.
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Dmitry Kleinbock, George Tomanov, Flows on <var>S</var>-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comment. Math. Helv. 82 (2007), no. 3, pp. 519–581DOI 10.4171/CMH/102