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The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved  X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved  X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

豪斯多夫维数测量一个空间的局部大小时，会考虑到点之间距离的度量。考虑半径最大为 r 的球数 ''N'' (r) ，需要完全覆盖 ''X''。当 r 很小时，n (r)以1/''r'' 增长多项式。对于一个表现足够好的 ''X''，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时， N(r)增长为1/r^d 。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的关键边界时，它等于豪斯多夫维数。

豪斯多夫维数测量一个空间的局部大小时，会考虑到点之间距离（度量）。考虑半径最大为 r 的球数 ''N'' (r) ，需要完全覆盖 ''X''。当 r 很小时，n (r)以1/''r'' 的多项式增长。对于一个表现足够好的 ''X''，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时， N(r)增长为1/r^d 。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的临界边界时，它等于豪斯多夫维数。