scholarly journals A capacity approach to box and packing dimensions of projections of sets and exceptional directions

2020 ◽  
Author(s):  
Kenneth Falconer
Keyword(s):  
Packing Dimensions

scholarly journals Packing dimension and measure of homogeneous Cantor sets

2006 ◽  
Vol 74 (3) ◽  
pp. 443-448 ◽  
Author(s):  
H.K. Baek
Keyword(s):  
Lower Bound ◽  
Explicit Formula ◽  
Cantor Sets ◽  
Packing Dimension ◽  
Packing Measure ◽  
Hausdorff Measures ◽  
Packing Dimensions

For a class of homogeneous Cantor sets, we find an explicit formula for their packing dimensions. We then turn our attention to the value of packing measures. The exact value of packing measure for homogeneous Cantor sets has not yet been calculated even though that of Hausdorff measures was evaluated by Qu, Rao and Su in (2001). We give a reasonable lower bound for the packing measures of homogeneous Cantor sets. Our results indicate that duality does not hold between Hausdorff and packing measures.

Projection theorems for box and packing dimensions

1996 ◽  
Vol 119 (2) ◽  
pp. 287-295 ◽  
Author(s):  
K. J. Falconer ◽  
J. D. Howroyd
Keyword(s):  
Orthogonal Projection ◽  
Packing Dimension ◽  
Analytic Subset ◽  
Box Counting ◽  
Packing Dimensions ◽  
Image Position ◽  
Almost All ◽  
Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

Packing dimensions of sections of sets

1999 ◽  
Vol 125 (1) ◽  
pp. 89-104 ◽  
Author(s):  
K. J. FALCONER ◽  
M. JÄRVENPÄÄ
Keyword(s):  
Packing Dimension ◽  
Packing Dimensions ◽  
Local Projections

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

Strong and weak duality principles for fractal dimension in Euclidean space

1995 ◽  
Vol 118 (3) ◽  
pp. 393-410 ◽  
Author(s):  
Colleen D. Cutler
Keyword(s):  
Strong Duality ◽  
The Other ◽  
Duality Principle ◽  
Weak Duality ◽  
Pointwise Dimension ◽  
Analytic Subset ◽  
Analytic Set ◽  
Borel Measures ◽  
Dual Representations ◽  
Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

Packing dimensions of homogeneous perfect sets

2007 ◽  
Vol 118 (1-2) ◽  
pp. 29-39 ◽  
Author(s):  
X. -Y. Wang ◽  
J. Wu
Keyword(s):  
Packing Dimensions

scholarly journals Dimension of the Boundary in Different Metrics

2013 ◽  
Vol 112 (2) ◽  
pp. 275
Author(s):  
Riku Klén ◽  
Ville Suomala
Keyword(s):  
Packing Dimensions ◽  
Euclidean Domains

We consider metrics on Euclidean domains $\Omega\subset\mathbf{R}^n$ that are induced by continuous densities $\rho\colon\Omega\rightarrow(0,\infty)$ and study the Hausdorff and packing dimensions of the boundary of $\Omega$ with respect to these metrics.

Hausdorff and packing dimensions and sections of measures

Mathematika ◽  
1998 ◽  
Vol 45 (1) ◽  
pp. 55-77 ◽  
Author(s):  
Maarit Järvenpää ◽  
Pertti Mattila
Keyword(s):  
Packing Dimensions

Fractal properties of products and projections of measures in ℝd

1994 ◽  
Vol 115 (3) ◽  
pp. 527-544 ◽  
Author(s):  
Xiaoyu Hu ◽  
S. James Taylor
Keyword(s):  
Dimensional Subspace ◽  
Fractal Measure ◽  
Borel Measures ◽  
Fractal Properties ◽  
Packing Dimensions ◽  
Fractal Measures ◽  
Almost All ◽  
Lower Dimensional

AbstractBorel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

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