Singular points of Hölder asymptotically optimally doubling measures

We consider the question of how the doubling characteristic of a measure determines the regularity of its support. The question was considered by David, Kenig, and Toro for codimension-1 under a crucial assumption of flatness, and later by Preiss, Tolsa, and Toro in higher codimension. However, their studies leave open the geometry of the support of such measures in a neighborhood about a non-flat point of the support. We here answer the question (in an almost classical sense) for codimension-1 Hölder doubling measures in $\RR^4$.

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