We develop a randomized algorithm (that succeeds with high probability) for generating an \(\epsilon \)-net in a sphere of dimension n. The basic scheme is to pick an alphabet consisting of \(O (n\ln (1/\epsilon )+\ln (1/\delta ))\) random rotations, form all possible words of length \(O (n\ln (1/\epsilon ))\) from this alphabet, and require these words act on a fixed point. We show the set of points so generated is equidistributed at a scale of \(\epsilon \).

中文翻译：

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使用随机旋转在球体上生成等分布网络

我们开发了一种随机算法（成功概率很高），用于在维度为n的球体中生成\(\epsilon \) -net  。基本方案是选择一个由\(O (n\ln (1/\epsilon )+\ln (1/\delta ))\)随机旋转组成的字母表，形成所有可能的长度为\(O (n\ ln (1/\epsilon ))\)来自这个字母表，并要求这些词作用于一个固定点。我们展示了这样生成的点集在\(\epsilon \)的尺度上是等分布的 。