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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ k$-dimensional regularity classifications for $ s$-fractals

Authors: Miguel Ángel Martín and Pertti Mattila
Journal: Trans. Amer. Math. Soc. 305 (1988), 293-315
MSC: Primary 28A75
MathSciNet review: 920160
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Abstract: We study subsets $ E$ of $ {{\mathbf{R}}^n}$ which are $ {H^s}$ measurable and have $ 0 < {H^s}(E) < \infty $, where $ {H^s}$ is the $ s$-dimensional Hausdorff measure. Given an integer $ k$, $ s \leqslant k \leqslant n$, we consider six ($ s$, $ k$) regularity definitions for $ E$ in terms of $ k$-dimensional subspaces or surfaces of $ {{\mathbf{R}}^n}$. If $ s = k$, they all agree with the ($ {H^k}$, $ k$) rectifiability in the sense of Federer, but in the case $ s < k$ we show that only two of them are equivalent. We also study sets with positive lower density, and projection properties in connection with these regularity definitions.

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Keywords: $ (s,\,k)$ regular sets, Hausdorff measures, tangent planes, orthogonal projections
Article copyright: © Copyright 1988 American Mathematical Society