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:

$\map {M_x} P = \displaystyle \sum_{k \mathop = 1}^\infty {r_k}^x$


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