Sharp conditions for equidistribution of translates of curves by a diagonal flow on $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(n,\mathbb{Z})$

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

 We consider the action of the diagonal subgroup $\{a(t)=(t^{n-1},t^{-1},\ldots,t^{-1})\}\subset G=\mathrm{SL}(n,\mathbb{R})$  on $X=G/\Gamma$, where $\Gamma=\mathrm{SL}(n,\mathbb{Z})$. Let $C$ be a finite piece of an analytic curve on the expanding horophere ($\cong{\mathbb R}^{n-1}$) of $\{a(t)\}_{{t>1}}$ in $G$ . Let $\mu_{C}$ be a smooth probability measure on the trajectory $C[\Gamma]$ 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 $a(t)\mu_{C}$ get equidistributed in $X$ as $t\to\infty$. 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.