奇异向量是欧式空间中满足特定丢番图逼近性质的向量，1920年代由Khintchine开始首先研究。杨鹏宇与Shah合作的工作中，给出了欧式空间仿射子空间中的奇异向量集的一个Hausdorff维数上界。技术上的创新点在于构造了齐性空间中新的Margulis高度函数；该函数对丢番图指数较小的点取值有限，且满足好的收缩性质。审稿人评价"It is the first paper I know which bounds the Hausdorff dimension of the set of singular objects (equivalently, points with divergent orbits) inside proper subsets of the ambient space. The method is natural (integral inequalities for Margulis functions) but the application to proper affine subspaces is quite original."

Publication:

Trans. Amer. Math. Soc. Ser. B 11 (2024), 1249-1265

https://doi.org/10.1090/btran/212

Author:

NIMISH A. SHAH

Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 

Email address: shah@math.osu.edu

PENGYU YANG

Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China Email address: yangpengyu@amss.ac.cn