Abstract
Let \({\mathscr {S}}\) be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of \({\mathscr {S}}\) is less than \(Kn^3\) for some \(K=o(n^{{1}/{7}})\) then, for n sufficiently large, all but at most O(K) points of \({\mathscr {S}}\) are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size n of an elliptic curve that span less than \(n^3/6\) solids containing exactly four points of \({\mathscr {S}}\).
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Acknowledgements
The authors would like to thank Massimo Giulietti for some useful discussions about lifts of elliptic curves. We also thank the referee whose detailed comments and suggestions were greatly appreciated.
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The first author acknowledges the support of the project MTM2017-82166-P of the Spanish Ministerio de Economía y Competitividad
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Ball, S., Jimenez, E. On Sets Defining Few Ordinary Solids. Discrete Comput Geom 66, 68–91 (2021). https://doi.org/10.1007/s00454-021-00302-7
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DOI: https://doi.org/10.1007/s00454-021-00302-7