On sets defining few ordinary planes
Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) poi...
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| Format: | article |
| Publication Date: | 2017 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/111528 |
| Online Access: | https://hdl.handle.net/2117/111528 https://dx.doi.org/10.1007/s00454-017-9935-2 |
| Access Level: | Open access |
| Keyword: | Geometry, Algebraic. Eight associated points theorem Green–Tao Ordinary planes Sylvester–Gallai Geometria Classificació AMS::51 Geometry::51M Real and complex geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria algebraica |
| Summary: | Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than (Formula presented.) then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than (Formula presented.) for some (Formula presented.) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four. |
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