ON SIMULTANEOUS LOCAL DIMENSION FUNCTIONS OF SUBSETS OF Rd
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초록
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.
Abstract
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.
참고문헌 (5)
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- L. Olsen, "Applications of divergence points to local dimension functions of subsets of Rd", Proc. Edinb. Math. Soc, 48(1), pp.213~218 (2005)
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- "Real Anal. Exchange", 1998, 373~389