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For a subset E ⊆ Rd and x ∈ Rd, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by dimH,loc(x,E) = lim r↘0 dimH(E ∩ B(x, r)) , dimP,loc(x,E) = lim r↘0 dimP(E ∩ B(x, r)) , where dimH and dimP denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions f, g : Rd → [0, d] with f ≤ g, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.
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참고문헌 (5)
참고문헌 신청
D. Spear / 1998 / Sets with different dimensions in [0, 1] / Real Anal. Exchange 24 (1) : 373 ~ 389
H. J¨urgensen / 1995 / Local Hausdorff dimension / Acta Inform 32 (5) : 491 ~ 507
K. J. Falconer / 1990 / Fractal Geometry / John Wiley & Sons
L. Olsen / 2005 / Applications of divergence points to local dimension functions of subsets of Rd / Proc. Edinb. Math. Soc 48 (1) : 213 ~ 218
T. Rushing / 1992 / Hausdorff dimension of wild fractals / Trans. Amer. Math. Soc 334 (2) : 597 ~ 613
H. J¨urgensen / 1995 / Local Hausdorff dimension / Acta Inform 32 (5) : 491 ~ 507
K. J. Falconer / 1990 / Fractal Geometry / John Wiley & Sons
L. Olsen / 2005 / Applications of divergence points to local dimension functions of subsets of Rd / Proc. Edinb. Math. Soc 48 (1) : 213 ~ 218
T. Rushing / 1992 / Hausdorff dimension of wild fractals / Trans. Amer. Math. Soc 334 (2) : 597 ~ 613
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