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Generating an equidistributed net on a sphere using random rotations. (English) Zbl 07457128

Summary: We develop a randomized algorithm (that succeeds with high probability) for generating an \(\varepsilon \)-net in a sphere of dimension \(n\). The basic scheme is to pick an alphabet consisting of \(O (n\ln (1/\varepsilon )+\ln (1/\delta ))\) random rotations, form all possible words of length \(O (n\ln (1/\varepsilon ))\) 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 \(\varepsilon \).

MSC:

68W20 Randomized algorithms
52C10 Erdős problems and related topics of discrete geometry
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