Sharp conditions for equidistribution of translates of curves by a diagonal flow on SL(n,R)/SL(n,Z)

Seminar: 
Group Actions and Dynamics
Event time: 
Monday, October 4, 2021 - 4:00pm
Location: 
Zoom
Speaker: 
Nimish Shah
Speaker affiliation: 
The Ohio State University
Event description: 

 We consider the action of the diagonal subgroup {a(t)=(tn1,t1,,t1)}G=SL(n,R)  on X=G/Γ, where Γ=SL(n,Z). Let C be a finite piece of an analytic curve on the expanding horophere (Rn1) of {a(t)}t>1 in G . Let μC be a smooth probability measure on the trajectory C[Γ] on X. We provide necessary and sufficient conditions on the smallest affine subspace containing C in terms of Diophantine approximation and algebraic number fields so that the measures a(t)μC get equidistributed in X as t. This result generalizes the speaker’s earlier work showing equidistribution of translates of curves, which are 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 D. Kleinbock, N.  de Saxcé, and P. Yang; and the general case is a joint work with Pengyu Yang.