## Abstract

We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a truly d-dimensional configuration of points v_{1}, . . . , v_{n} ε ℂ^{d}. More formally, we show that, if the number of r-rich lines is significantly larger than n^{2}/r^{d} then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^{d} can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemerédi-Trotter theorem which gives a bound of n_{2}/r_{3} on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R3 in the seminal work of Guth and Katz [7] and was also recently proved over R4 (under some additional restrictions) [14]. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r - 2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of [1] to convert these local linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.

Original language | English (US) |
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Title of host publication | 31st International Symposium on Computational Geometry, SoCG 2015 |

Editors | Janos Pach, Janos Pach, Lars Arge |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 584-598 |

Number of pages | 15 |

ISBN (Electronic) | 9783939897835 |

DOIs | |

State | Published - Jun 1 2015 |

Event | 31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands Duration: Jun 22 2015 → Jun 25 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 34 |

ISSN (Print) | 1868-8969 |

### Other

Other | 31st International Symposium on Computational Geometry, SoCG 2015 |
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Country | Netherlands |

City | Eindhoven |

Period | 6/22/15 → 6/25/15 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- Additive Combinatorics
- Combinatorial Geometry
- Designs
- Incidences
- Polynomial Method