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If $S$ is a set of an countably infinite number of points uniformly distributed throughout the unit ball in $\mathbb R^n$, is there for every point $p$ in the ball and every real number $e>0$, a point $p_k\in S$ such that the distance $|p-p_k|, and is there an infinite number of such points $p_k$? Is it true for all $n>0$? What if $n=+\infty$?

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    As @anon said, the case $n$ finite is clear. When $n=+\infty$, you should explain what you mean by *uniformly distributed throughout the unit ball in $\mathbb{R}^n$*.2011-07-12

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