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|
uniform distribution on unit ball
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general-topology
probability-distributions
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0First of all, if the set is countable then there can't be an uncountable number of such points. I think you mean plain-jane infinite. – 2011-07-12
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1You should probably stipulate that the points are independent. If S were countable, then there would be a countable number of such points $p_k$, by the Second Borel-Cantelli Lemma. – 2011-07-12
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1I've never heard uniform distribution refer to a set rather than a random variable, but if we take it to mean the limiting density of points in any measureable region is simply equal to the region's volume divided by the ball's, then (fairly trivially) there must necessarily be an infinite number of points within any arbitrarily small region of the ball, hence answering your question in the affirmative. – 2011-07-12
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0As @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