@article{Li2020, 
author = {Nan Li and Aaron Naber},
title = {Quantitative Estimates on the Singular Sets of Alexandrov Spaces},
year = {2020},
journal = {Peking Mathematical Journal},
volume = {3},
number = {2},
pages = {203-234},
keywords = {Quantitative analysis, Singular sets, Alexandrov spaces, Sectional curvature, Splitting},
url = {https://www.sciopen.com/article/10.1007/s42543-020-00026-2},
doi = {10.1007/s42543-020-00026-2},
abstract = {Let  X∈Alexn(−1) be an n-dimensional Alexandrov space with curvature  ≥−1. Let the r-scale  (k,ϵ)-singular set  Sϵ,rk(X) be the collection of  x∈X so that  Br(x) is not  ϵr-close to a ball in any splitting space  Rk+1×Z. We show that there exists  C(n,ϵ)>0 and  β(n,ϵ)>0, independent of the volume, so that for any disjoint collection  {Bri(xi):xi∈Sϵ,βrik(X)∩B1,ri≤1}, the packing estimate  ∑rik≤C holds. Consequently, we obtain the Hausdorff measure estimates  Hk(Sϵk(X)∩B1)≤C and  Hn(Br(Sϵ,rk(X))∩B1(p))≤Crn−k. This answers an open question in Kapovitch et al. (Metric-measure boundary and geodesic flow on Alexandrov spaces. arXiv: 1705.04767 (2017)). We also show that the k-singular set  Sk(X)=⋃ϵ>0(⋂r>0Sϵ,rk) is k-rectifiable and construct examples to show that such a structure is sharp. For instance, in the  k=1 case we can build for any closed set  T⊆S1 and  ϵ>0 a space  Y∈Alex3(0) with  Sϵ1(Y)=ϕ(T), where  ϕ:S1→Y is a bi-Lipschitz embedding. Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable, 1-Cantor set with positive 1-Hausdorff measure.}
}