Equidistribution of expanding translates of curves and Diophantine approximation on matrices

Pengyu Yang ETH Zurich

Dynamical Systems Number Theory mathscidoc:2105.11001

Inventiones Mathematicae, 220, 909-948, 2020.1
We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, 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, 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.
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@inproceedings{pengyu2020equidistribution,
  title={Equidistribution of expanding translates of curves and Diophantine approximation on matrices},
  author={Pengyu Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210527033659066641838},
  booktitle={Inventiones Mathematicae},
  volume={220},
  pages={909-948},
  year={2020},
}
Pengyu Yang. Equidistribution of expanding translates of curves and Diophantine approximation on matrices. 2020. Vol. 220. In Inventiones Mathematicae. pp.909-948. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210527033659066641838.
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