Hausdorff dimension - Knowlpedia

For a set $E$ in a metric space, its Hausdorff dimension is

\dim_H E=\inf\{s>0:\mathcal H^s(E)=0\},

with infimum infinity if this set is empty. Here $\mathcal H^s$ is Hausdorff measure for the specified metric. The definition assigns dimension zero to the empty set; some authors use a different empty-set convention.

Critical exponent

If $s<t$ , a cover by sets of diameter at most $\delta$ has its $t$ -cost bounded by $\delta^{t-s}$ times its $s$ -cost. Hence finiteness at exponent $s$ forces vanishing at each larger exponent. At the critical exponent, the measure can be zero, finite positive, or infinite.