= Mass distribution principle
{title2=$\mu(B(x,r))\leq Cr^s\ \Longrightarrow\ \dim_HE\geq s$}
The mass distribution principle converts a diffuse <measure> into a lower bound on <Hausdorff dimension>. Let $s>0$ and let a <Borel probability measure> $\mu$ give a <Borel set> $E$ full mass. If, for some $C,r_0>0$, every <ball> centered in $E$ satisfies
$$
\mu(B(x,r))\leq Cr^s\qquad(0<r<r_0),
$$
then $\mathcal H^s(E)\geq1/C>0$ in the diameter-cost convention, and consequently $\dim_HE\geq s$.
The idea is to keep unit mass from being crowded into cheap covering pieces. For a piece $U$ of positive <diameter> $r$ meeting $E$, choose $x\in U\cap E$. A ball of radius $r+\varepsilon$ contains $U$; letting $\varepsilon\downarrow0$ gives $\mu^*(U)\leq Cr^s$, where $\mu^*$ is the associated <outer measure>. Zero-diameter pieces have zero mass by the same estimate. For any sufficiently small <cover>,
$$
1=\mu(E)\leq\sum_j\mu^*(U_j)\leq C\sum_j(\operatorname{diam}U_j)^s.
$$
Thus every cover costs at least $1/C$. This is the lower-bound counterpart to the explicit covers in <scale-restricted Hausdorff content>. The <Cantor distribution> supplies the needed mass in <Hausdorff dimension of the Cantor set>. https://www.math.stonybrook.edu/~bishop/fractalbook.pdf[Bishop and Peres, §1.2: the mass distribution principle]{ref}.
Back to article page
