Ahlfors–David regular set

Definition

Let $0\le\delta\le d$ , $C\ge1$ , and $0<\alpha_0<\alpha_1$ . A set $X\subseteq\mathbb R^d$ is Ahlfors–David $\delta$ -regular with constant $C$ from scales $\alpha_0$ to $\alpha_1$ if there is a measure $\mu$ supported on $X$ such that every ball $B$ of diameter $\alpha_0<R<\alpha_1$ satisfies

\mu(B)\le C R^\delta,

and, whenever the center of $B$ lies in $X$ , also

\mu(B)\ge C^{-1}R^\delta.

Interpretation

The exponent $\delta$ is a uniform effective dimension throughout the specified scale range. The upper bound prevents excessive concentration; the centered lower bound prevents large holes relative to the support.

Relation to porosity

On the real line, an Ahlfors–David regular set of dimension $\delta<1$ is porous, with constants depending on $C$ and $\delta$ . Conversely, a truncated porous subset of $\mathbb R$ can be enlarged to a regular set of some dimension below one. In higher dimensions, regularity and line porosity are distinct.

References

Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995. DOI record.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: §1.3.