Quantitative Estimates on the Singular Sets of Alexandrov Spaces

Let $X\in \text{Alex}{}^n(-1)$ be an n-dimensional Alexandrov space with curvature $\ge -1$. Let the r-scale $(k,ϵ)$-singular set ${\mathcal{S}}_{ϵ,r}^k(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $ϵr$-close to a ball in any splitting space ${\mathbb{R}}^{k+1}\times Z$. We show that there exists $C(n,ϵ)>0$ and $\beta (n,ϵ)>0$, independent of the volume, so that for any disjoint collection $\{B_{r_i}(x_i):x_i\in {\mathcal{S}}_{ϵ,\beta r_i}^k(X)\cap B_1,r_i\le 1\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates ${\mathcal{H}}^k({\mathcal{S}}_{ϵ}^k(X)\cap B_1)\le C$ and ${\mathcal{H}}^n(B_r({\mathcal{S}}_{ϵ,r}^k(X))\cap B_1(p))\le C{r}^{n-k}$. 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 ${\mathcal{S}}^k(X)=\bigcup _{ϵ>0}(\bigcap _{r>0}{\mathcal{S}}_{ϵ,r}^k)$ 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 $T\subseteq {\mathbb{S}}^1$ and $ϵ>0$ a space $Y\in {\text{Alex}}^3(0)$ with ${\mathcal{S}}_{ϵ}^1(Y)=\varphi (T)$, where $\varphi :{\mathbb{S}}^1\to Y$ 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.