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

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

2005-06-24

arxiv.org/abs/math/0506510v1

Mathematics

Number Theory

56 pages; an earlier version is available as an MPI (Bonn) preprint, 2003

Scientific paper

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 $\Bbb R^n$) are generalized to the $S$-arithmetic setting.

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