Hausdorff dimension

Let $\mathrm{\Theta }$ be a bounded subset of ${ℝ}^n$let $N_{\mathrm{\Theta }}(ϵ)$ be the minimum number of balls of radius $ϵ$ required to cover $\mathrm{\Theta }$. Then define the Hausdorff dimension$d_H$ of $\mathrm{\Theta }$ to be

$$d_H(\mathrm{\Theta }):=-\underset{ϵ\to 0}{lim}\frac{\log N_{\mathrm{\Theta }}(ϵ)}{\log ϵ}.$$

Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve. Each of these may be covered with a collection of scaled-down copies of itself. In fact, in the case of the Sierpinski gasket, one can take the individual triangles in each approximation as balls in the covering. At stage $n$, there are $3^n$ triangles of radius $\frac{1}{2^n}$, and so the Hausdorff dimension of the Sierpinski triangle is at most $-\frac{n\log 3}{n\log 1/2}=\frac{\log 3}{\log 2}$, and it can be shown that it is equal to $\frac{\log 3}{\log 2}$.

From some notes from Koro

This definition can be extended to a general metric space $X$ with distance function $d$.

Define the diameter $|C|$ of a bounded subset $C$ of $X$ to be ${sup}_{x,y\in C}d(x,y)$.

Define a $r$-coverof $X$ to be a collection of subsets $C_i$ of $X$ indexed by some countable set $I$, such that and $X={\cup }_{i\in I}{C}_i$.

We also define the function

$$H_r^D(X)=inf\sum _{i\in I}{|C_i|}^D$$

where the infimum is over all countable $r$-covers of $X$.The Hausdorff dimension of $X$ may then be defined as

$$d_H(X)=inf\{D\mid \underset{r\to 0}{lim}{H}_r^D(X)=0\}.$$

When $X$ is a subset of ${ℝ}^n$ with any norm-induced metric, then this definition reduces to that given above.