We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space G/Γ of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety G/P , which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved.The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.
Let L be a simply-connected simple connected algebraic group over a number field F , and H be a semisimple absolutely maximal connected F -subgroup of L. Under a cohomological condition, we prove an asymptotic formula for the number of rational points of bounded height on projective equivariant compactifications of ∆(H)\L n with respect to a balanced line bundle, where ∆(H) is the image of H diagonally embedded in L n .
For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds.We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.
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