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Transactions of the American Mathematical Society

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$ 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.


References [Enhancements On Off] (What's this?)

  • [FK] K. J. Falconer, Geometry of fractal sets, Cambridge Univ. Press, 1985. MR 867284 (88d:28001)
  • [FH] H. Federer, Geometric measure theory, Springer-Verlag, 1969. MR 0257325 (41:1976)
  • [G] M. de Guzmán, Differentiation of integrals in $ {R^n}$, Springer-Verlag, 1975.
  • [H] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [MB] B. B. Mandelbrot, The fractal geometry of nature, Freeman, 1982. MR 665254 (84h:00021)
  • [MM] M. A. Martin, Propiedades de proyección de fraciales, Tesis Doctoral, Universidad Complutense, Madrid, 1986.
  • [MJ] J. M. Marstrand, The ($ \phi $, $ s$) regular subsets of $ n$-space, Trans. Amer. Math. Soc. 113 (1964), 369-392. MR 0166336 (29:3613)
  • [MP] P. Mattila, On the structure of self-similar fractals, Ann. Acad. Sci. Fenn. Ser. A I 7 (1982), 189-195. MR 686639 (84j:28011)
  • [P] D. Preiss, Geometry of measures in $ {R^n}$, distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. MR 890162 (88d:28008)
  • [R] M. Ross, Federer's structure theorem, Research Report, Centre for Mathematical Analysis, Australian National University, 1984.
  • [S] A. Salli, Upper density theorems for Hausdorff measures on fractals, Dissertationes Ann. Acad. Sci. Fenn. Ser. A I 55 (1985).
  • [W] B. White, Problem 3.10, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc, Providence, R. I., 1985, p. 447.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0920160-2
Keywords: $ (s,\,k)$ regular sets, Hausdorff measures, tangent planes, orthogonal projections
Article copyright: © Copyright 1988 American Mathematical Society

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