= Hausdorff dimension of the Cantor set
{c}
{title2=$\dim_HC=\log2/\log3$}
The middle-third <Cantor set> has <Hausdorff dimension>
$$
d=\frac{\log2}{\log3}\approx0.63093.
$$
Its <Lebesgue measure> is zero, yet $0<\mathcal H^d(C)<\infty$. These assertions measure different aspects of size: interval <length> is too coarse to distinguish this set from a <null set>, while the critical fractional measuring rule captures its remaining structure.
At level $n$, the construction <covers> $C$ by $2^n$ intervals of <diameter> $3^{-n}$. Their total $s$-cost is
$$
2^n(3^{-n})^s=(2\,3^{-s})^n.
$$
For $s>d$ it tends to zero, proving the upper dimension bound. At $s=d$ it equals one, giving $\mathcal H^d(C)\leq1$. For $s<d$ this particular <cover>'s cost grows, but that alone would not prove a lower bound on <Hausdorff measure>.
\Image[/hausdorff-cover-cost-threshold.png]
{title=Cantor cover costs change trend at the critical exponent, and distributed mass supplies the lower bound}
For the matching lower bound, use the <Cantor distribution>, which assigns mass $2^{-n}$ to each level-$n$ <Cantor cylinder>. An interval of length $r$ with $3^{-(n+1)}<r\leq3^{-n}$ meets at most two level-$n$ construction intervals, since their gaps are at least $3^{-n}$. Its mass is therefore at most
$$
2\,2^{-n}=2(3^{-n})^d<2\,3^d r^d=4r^d.
$$
Every covering <set> of <diameter> $r$ in the line lies in an interval of length $r$, so any small <cover> must have total $d$-cost at least $1/4$. This is the <mass distribution principle> in its interval form: $1/4\leq\mathcal H^d(C)\leq1$, sufficient to prove the claimed dimension. The constants are convenient bounds, rather than an assertion of the exact critical <Hausdorff measure>. In the diagram, the right panel is a finite-stage approximation to the <Cantor function>, showing how probability is shared while the middle third receives none.
The reusable pattern is to obtain an upper bound by constructing <covers> and a lower bound by distributing mass. The same strategy underlies <Cartesian powers of the Cantor set> and the dimension formula for a <finite similarity iterated function system> satisfying the <open set condition>.
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