Packing dimension

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In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982.

Definitions[edit]

Let (Xd) be a metric space with a subset S ⊆ X and let s ≥ 0. The s-dimensional packing pre-measure of S is defined to be

P_0^s (S) = \lim_{\delta \downarrow 0} \sup \left\{ \left. \sum_{i \in I} \mathrm{diam} (B_i)^s \right| \begin{matrix} \{ B_i \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}.

Unfortunately, this is just a pre-measure and not a true measure on subsets of X, as can be seen by considering dense, countable subsets. However, the pre-measure leads to a bona fide measure: the s-dimensional packing measure of S is defined to be

P^s (S) = \inf \left\{ \left. \sum_{j \in J} P_0^s (S_j) \right| S \subseteq \bigcup_{j \in J} S_j, J \text{ countable} \right\},

i.e., the packing measure of S is the infimum of the packing pre-measures of countable covers of S.

Having done this, the packing dimension dimP(S) of S is defined analogously to the Hausdorff dimension:

\begin{align}
\dim_{\mathrm{P}} (S) &{}  = \sup \{ s \geq 0 | P^s (S) = + \infty \} \\
&{} = \inf \{ s \geq 0 | P^s (S) = 0 \}.
\end{align}

An example[edit]

The following example is the simplest situation where Hausdorff and packing dimensions may differ.

Fix a sequence (a_n) such that a_0=1 and 0<a_{n+1}<a_n/2. Define inductively a nested sequence E_0 \supset E_1 \supset E_2 \supset \cdots of compact subsets of the real line as follows: Let E_0=[0,1]. For each connected component of E_n (which will necessarily be an interval of length a_n), delete the middle interval of length a_n - 2a_{n+1}, obtaining two intervals of length a_{n+1}, which will be taken as connected components of E_{n+1}. Next, define K = \bigcap_n E_n. Then K is topologically a Cantor set (i.e., a compact totally disconnected perfect space). For example, K will be the usual middle-thirds Cantor set if a_n=3^{-n}.

It is possible to show that the Hausdorff and the packing dimensions of the set K are given respectively by:

\begin{align}
\dim_{\mathrm{H}} (K) &{}  = \liminf_{n\to\infty} \frac{n \log 2}{- \log a_n} \, , \\
\dim_{\mathrm{P}} (K) &{}  = \limsup_{n\to\infty} \frac{n \log 2}{- \log a_n} \, .
\end{align}

It follows easily that given numbers 0 \leq d_1 \leq d_2 \leq 1, one can choose a sequence (a_n) as above such that the associated (topological) Cantor set K has Hausdorff dimension d_1 and packing dimension d_2.

Generalizations[edit]

One can consider dimension functions more general than "diameter to the s": for any function h : [0, +∞) → [0, +∞], let the packing pre-measure of S with dimension function h be given by

P_0^h (S) = \lim_{\delta \downarrow 0} \sup \left\{ \left. \sum_{i \in I} h \big( \mathrm{diam} (B_i) \big) \right| \begin{matrix} \{ B_{i} \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}

and define the packing measure of S with dimension function h by

P^h (S) = \inf \left\{ \left. \sum_{j \in J} P_0^h (S_j) \right| S \subseteq \bigcup_{j \in J} S_j, J \text{ countable} \right\}.

The function h is said to be an exact (packing) dimension function for S if Ph(S) is both finite and strictly positive.

Properties[edit]

  • If S is a subset of n-dimensional Euclidean space Rn with its usual metric, then the packing dimension of S is equal to the upper modified box dimension of S:
\dim_{\mathrm{P}} (S) = \overline{\dim}_\mathrm{MB} (S).
This result is interesting because it shows how a dimension derived from a measure (packing dimension) agrees with one derived without using a measure (the modified box dimension).

Note, however, that the packing dimension is not equal to the box dimension. For example, the set of rationals Q has box dimension one and packing dimension zero.

See also[edit]

References[edit]