Packing dimension

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

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

It is possible to show that the Hausdorff and the packing dimensions of the set ${\displaystyle K}$ are given respectively by:

${\displaystyle {\begin{aligned}\dim _{\mathrm {H} }(K)&{}=\liminf _{n\to \infty }{\frac {n\log 2}{-\log a_{n}}}\,,\\\dim _{\mathrm {P} }(K)&{}=\limsup _{n\to \infty }{\frac {n\log 2}{-\log a_{n}}}\,.\end{aligned}}}$

It follows easily that given numbers ${\displaystyle 0\leq d_{1}\leq d_{2}\leq 1}$, one can choose a sequence ${\displaystyle (a_{n})}$ as above such that the associated (topological) Cantor set ${\displaystyle K}$ has Hausdorff dimension ${\displaystyle d_{1}}$ and packing dimension ${\displaystyle d_{2}}$.

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

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

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

${\displaystyle P^{h}(S)=\inf \left\{\left.\sum _{j\in J}P_{0}^{h}(S_{j})\right|S\subseteq \bigcup _{j\in J}S_{j},J{\text{ countable}}\right\}.}$

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