Sharp conditions for equidistribution of translates of a subspace of a horosphere by a diagonal flow in the space of unimodular lattices

Abstract: We consider the action of the one-parameter subgroup $a(t) = \text{exp}((n-1)t, -t, \ldots, -t)$ of SL(n,R) on the space X of unimodular lattices in $R^n$. Let C be an analytic curve on the expanding horosphere of a(t) in X through the standard lattice $Z^n$. Let μ be a smooth probability measure on C. We describe necessary and sufficient conditions, in terms of Diophantine approximation and algebraic number fields, on the smallest affine subspace containing C so that the translated measures a(t)μ get equidistributed in X as $t \rightarrow \infty$. This generalizes my earlier result showing equidistribution of translates of curves not contained in proper affine subspaces. The result answers a question of Davenport and Schmidt on non-improvability of Dirichlet’s approximation. [The case of n = 3 is a joint work with D. Kleinbock, N. DeSaxe, and P. Yang; and the general case is a joint work with P. Yang.]

algebraic geometrydifferential geometrygroup theorygeometric topologynumber theory

Audience: general audience

International Conference on Discrete groups, Geometry and Arithmetic

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