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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0920160-2
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: <IMG WIDTH="54" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$(s,\,k)$"> regular sets, Hausdorff measures, tangent planes, orthogonal projections
Article copyright: © Copyright 1988 American Mathematical Society