MathDB

Let

\{A_n | n = 1, 2, \cdots \}

be a set of points in the plane such that for each

n

, the disk with center

A_n

and radius

2^n

contains no other point

A_j

. For any given positive real numbers

a < b

and

R

, show that there is a subset

G

of the plane satisfying:
(i) the area of

G

is greater than or equal to

R

;
(ii) for each point

P

G

a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.

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