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# Hausdorff dimension

Let $\Theta$ be a bounded^{} subset of $\mathbb{R}^{n}$
let $N_{{\Theta}}(\epsilon)$ be the minimum number of balls of radius $\epsilon$ required to cover $\Theta$. Then define the *Hausdorff dimension*
$d_{H}$ of $\Theta$ to be

$d_{H}(\Theta):=-\lim_{{\epsilon\rightarrow 0}}\frac{\log N_{{\Theta}}(\epsilon% )}{\log\epsilon}.$ |

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 *countable $r$-cover*
of $X$ to be a collection of subsets $C_{i}$ of $X$ indexed by some countable set $I$, such that $|C_{i}|<r$ and $X=\cup_{{i\in I}}C_{i}$.

We also define the function

$H^{D}_{r}(X)=\inf\sum_{{i\in I}}|C_{i}|^{D}$ |

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

$d_{H}(X)=\inf\{D\mid\lim_{{r\rightarrow 0}}H^{D}_{r}(X)=0\}.$ |

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

## Mathematics Subject Classification

28A80*no label found*

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## Comments

## Hausdorff dimension

The definition at the top d_H(\Theta) is NOT Hausdorff dimension. It is "box dimension" or "Bouligand dimension".

The "Notes from Koro" provide the correct definition.

Even in Euclidean space, the two values can differ. For example, the rational numbers in the interval [0,1] ... box dimension 1 but Hausdorff dimension 0.