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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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$k$-dimensional regularity classifications for $s$-fractals
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by Miguel Ángel Martín and Pertti Mattila PDF
Trans. Amer. Math. Soc. 305 (1988), 293-315 Request permission

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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Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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