Abstract
We show that a Cantor set in RN must have two points whose tangent cones are not isomorphic, in the sense of being related by a linear transformation. We define two integer invariants measuring the size of a tangent cone, lineality and the new concept of density, and prove that there are two cones having different lineality or density. As a consequence, a Cantor set in RN cannot be differentiably homogeneous. The latter result fills a gap between previous results showing that Cantor sets may be Lipschitz ambient homogeneous but not C1 homogeneous.
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