Packing dimension

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.
Let ("X", "d") be a metric space with a subset "S" ⊆ "X" and let "s" ≥ 0 be a real number. The "s"-dimensional packing pre-measure of "S" is defined to be
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
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:
An example.
The following example is the simplest situation where Hausdorff and packing dimensions may differ.
Fix a sequence formula_4 such that formula_5 and formula_6. Define inductively a nested sequence formula_7 of compact subsets of the real line as follows: Let formula_8. For each connected component of formula_9 (which will necessarily be an interval of length formula_10), delete the middle interval of length formula_11, obtaining two intervals of length formula_12, which will be taken as connected components of formula_13. Next, define formula_14. Then formula_15 is topologically a Cantor set (i.e., a compact totally disconnected perfect space). For example, formula_15 will be the usual middle-thirds Cantor set if formula_17.
It is possible to show that the Hausdorff and the packing dimensions of the set formula_15 are given respectively by:
It follows easily that given numbers formula_20, one can choose a sequence formula_4 as above such that the associated (topological) Cantor set formula_15 has Hausdorff dimension formula_23 and packing dimension formula_24.
Generalizations.
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
and define the packing measure of "S" with dimension function "h" by
The function "h" is said to be an exact (packing) dimension function for "S" if "P""h"("S") is both finite and strictly positive.
Properties.
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.