My research focuses on dynamical systems of group actions on homogeneous spaces and their applications to number theory. Recently I am interested in the following topics:
An effective version of the Oppenheim conjecture with a polynomial error rate. Joint with Elon Lindenstrauss, Amir Mohammadi, and Zhiren Wang. arXiv:2305.18271
Effective version of Ratner’s equidistribution theorem for $\mathrm{SL}(3, \mathbb{R})$. arXiv:2208.02525
Effective equidistribution for multiplicative Diophantine approximation on lines. Joint with Sam Chow. Inventiones Mathematicae, Online, arXiv:1902.06081
Khintchine’s theorem and Diophantine approximation on manifolds. Joint with Victor Beresnevich. Acta Mathematica, Volume 231 (2023), no. 1, 1-30. arXiv:2105.13872
Winning property of badly approximable points on curves. Joint with Victor Beresnevich and Erez Nesharim. Duke Math. J. Volume 171 (2022), no. 14, 2841-2880. arXiv:2005.02128
Equidistribution of expanding translates of curves in homogeneous spaces with the action of $(\mathrm{SO}(n,1))^k$. Acta Math. Sin. (Engl. Ser.), Volume 38 (2022), no. 1, 250-224.
$\mathbf{Bad}(w)$ is hyperplane absolute winning. Joint with Victor Beresnevich and Erez Nesharim. Geometric and Functional Analysis, Volume 31 (2021), Issue 1, 1-33. arXiv:2005.11947
Equidistribution of curves in homogeneous spaces and Dirichlet’s approximation Theorem for matrices. Joint with Nimish Shah. Discrete and Continuous Dynamical Systems-A, Volume 40 (2020), no. 9, 5247-5287. arXiv:1606.00152
Badly approximable points on manifolds and unipotent orbits in homogeneous spaces. Geometric and Functional Analysis, Volume 29 (2019), Issue 4, 1194–1234. arXiv:1703.03461
Hausdorff dimension of divergent diagonal geodesics on product of finite volume hyperbolic spaces. Ergodic Theory and Dynamical Systems, volume 39 (2019), no. 5, 1401-1439. arXiv:1303.5993
Exponential mixing and smooth classification of commuting expanding maps. Joint with Ralf Spatzier. Journal of Modern Dynamics, volume 11 (2017), 263-312. arXiv:1608.02295
Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation for square matrices. Proc. Amer. Math. Soc. 144 (2016), no. 12, 5291-5308. arXiv:1311.1106
Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces. Israel J. Math. 216 (2016), no. 1, 389-413. arXiv:1303.6023