Definition:Solid-Packing Constant for Circles
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Theorem
Let $P = \sequence {D_1, D_2, \dotsc}$ be an infinite sequence of disjoint open disks whose union is the unit disk $D$ except for a set of measure zero.
Let $r_n$ be the radius of $D_n$.
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $\map {M_x} P$ be defined as:
- $\map {M_x} P = \ds \sum_{k \mathop = 1}^\infty {r_k}^x$
For each $P$, there exists a (real) number $\map e P$ such that:
- $\map {M_x} P$ is divergent for $x < \map e P$
- $\map {M_x} P$ is convergent for $x > \map e P$
From the Mergelyan-Wesler Theorem:
- $1 < \map e P < 2$
for all $P$.
The constant $S$ such that:
- $S < \map e P$
is known as the solid-packing constant for circles.
It can be interpreted as the fractal dimension of the set of points of $P$ which are not covered by the $D_n$ open disks.
Sources
- 1969: Z.A. Melzak: On the solid packing constant for circles (Math. Comp. Vol. 23: pp. 169 – 172) www.jstor.org/stable/2005066
- 1977: Benoit B. Mandelbrot: Fractals: Form, Chance and Dimension
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,30695 1 \ldots$