Question 52
(SWE 5) In the plane a point $O$ and a sequence of points $P_1,P_2,P_3,\ldots$ are given. The distances $OP_1,OP_2,OP_3,\ldots$ are $r_1,r_2,r_3,\ldots$, where $r_1\le r_2\le r_3\le\cdots$. Let $\alpha$ satisfy $0\lt \alpha\lt 1$. Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is greater than or equal to $r_n^\alpha$. Determine the exponent $\beta$, as large as possible, such that for some $C$ independent of $n$,[2]
$$r_n\ge Cn^\beta,\qquad n=1,2,\ldots.$$
[2] This problem is not elementary. The solution offered by the proposer, which is not quite clear and complete, only shows that if such a $\beta$ exists, then $\beta\ge\frac1{2(1-\alpha)}$.
$$r_n\ge Cn^\beta,\qquad n=1,2,\ldots.$$
[2] This problem is not elementary. The solution offered by the proposer, which is not quite clear and complete, only shows that if such a $\beta$ exists, then $\beta\ge\frac1{2(1-\alpha)}$.