ON SIMULTANEOUS LOCAL DIMENSION FUNCTIONS OF SUBSETS OF ℝ^d

Abstract

For a subset $E{\subseteq}\mathbb{R}^d$ and $x{\in}\mathbb{R}^d$, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by $$dim_{H,loc}(x,E)=\lim_{r{\searrow}0}dim_H(E{\cap}B(x,r))$$, $$dim_{P,loc}(x,E)=\lim_{r{\searrow}0}dim_P(E{\cap}B(x,r))$$, where $dim_H$ and $dim_P$ 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 $f,g:\mathbb{R}^d{\rightarrow}[0,d]$ with $f{\leq}g$, 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.

Keywords

• Hausdorff dimension;
• packing dimension;
• local Hausdorff dimension;
• local packing dimension

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References

1. K. J. Falconer, Fractal Geometry, John Wiley & Sons, 1990.
2. H. Jurgensen and L. Staiger, Local Hausdorff dimension, Acta Inform. 32 (1995), no. 5, 491-507. https://doi.org/10.1007/BF01213081
3. L. Olsen, Applications of divergence points to local dimension functions of subsets of ${\mathbb{R}}^d$, Proc. Edinb. Math. Soc. 48 (2005), no. 1, 213-218. https://doi.org/10.1017/S0013091503000798
4. T. Rushing, Hausdorff dimension of wild fractals, Trans. Amer. Math. Soc. 334 (1992), no. 2, 597-613. https://doi.org/10.1090/S0002-9947-1992-1162104-8
5. D. Spear, Sets with different dimensions in [0, 1], Real Anal. Exchange 24 (1998/99), no. 1, 373-389.