Shrinking targets problems for flows on homogeneous spaces

We study shrinking targets problems for discrete time flows on a homogenous space $Γ\backslash G$ with $G$ a semisimple group and $Γ$ an irreducible lattice. Our results apply to both diagonalizable and unipotent flows, and apply to very general families of shrinking targets. As a special case, we establish logarithm laws for cusp excursions of unipotent flows answering a question of Athreya and Margulis.

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