Quantitative estimates on the singular sets of Alexandrov spaces

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Abstract: Let

X i n e x t A l e x,^{n} (- 1)

be an

n

-dimensional Alexandrov space with curvature

g e - 1

. Let the

r

-scale

(k, e p s i l o n)

-singular set

m a t h c a l S_{e p s i l o n,, r}^{k} (X)

be the collection of

x i n X

so that

B_{r} (x)

is not

e p s i l o n r

-close to a ball in any splitting space

m a t h b b R^{k + 1} i m e s Z

. We show that there exists

C (n, e p s i l o n) > 0

and

, independent of the volume, so that for any disjoint collection

, the packing estimate

s u m r_{i}^{k} l e C

holds. Consequently, we obtain the Hausdorff measure estimates

m a t h c a l H^{k} (m a t h c a l S_{e}^{k} p s i l o n (X) c a p B_{1}) l e C

and

. This answers an open question asked by Kapovitch and Lytchak. We also show that the

k

-singular set

m a t h c a l S^{k} (X) = u n d e r s e t e p s i l o n > 0 c u p l e f t (u n d e r s e t r > 0 c a p m a t h c a l S_{e p s i l o n,, r}^{k} i g h t)

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 s u b s e t e q m a t h b b S^{1}

and

e p s i l o n > 0

a space

Y i n e x t {A l e x}^{3} (0)

with

m a t h c a l S_{e}^{1} p s i l o n (Y) = p h i (T)

, where

p h i c o l o n m a t h b b S^{1} o 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.

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