Mergelyan-Wesler Theorem

Let $P=\{D_1, D_2, \ldots\}$ be an infinite set of disjoint open Disks $D_n$ of radius $r_n$ such that the union is almost the unit Disk. Then

$$\sum_{n=1}^\infty r_n=\infty.$$ (1)

Define

$$M_x(P)\equiv \sum_{n=1}^\infty {r_n}^x.$$ (2)

Then there is a number $e(P)$ such that $M_x(P)$ diverges for $x<e(P)$ and converges for $x>e(P)$. The above theorem gives

$$1<e(P)<2.$$ (3)

There exists a constant which improves the inequality, and the best value known is

$$S=1.306951\ldots.$$ (4)

References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 36-37, 1983.

Mandelbrot, B. B. Fractals. San Francisco, CA: W. H. Freeman, p. 187, 1977.

Melzack, Z. A. ``On the Solid Packing Constant for Circles.'' Math. Comput. 23, 1969.