Abstract
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 \(\epsilon \).
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Acknowledgements
The authors acknowledge support of DAE project no. 12-R&D-TFR-5.01-0500. HN was partially supported by a Ramanujan fellowship. SC would like to thank Sandeep Juneja and Jaikumar Radhakrishnan for helpful comments. We thank the anonymous referee for very detailed review and several helpful comments and suggestions that improved the presentation significantly.
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Chakraborty, S., Narayanan, H. Generating an Equidistributed Net on a Sphere Using Random Rotations. Discrete Comput Geom 67, 231–257 (2022). https://doi.org/10.1007/s00454-021-00333-0
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DOI: https://doi.org/10.1007/s00454-021-00333-0