## Abstract

Let L be a set of lines of an affine space over a field and let S be a set of points with the property that every line of L is incident with at least N points of S. Let D be the set of directions of the lines of L considered as points of the projective space at infinity. We give a geometric construction of a set of lines L, where D contains an Nn−1 grid and where S has size 2( 1 2N)n plus smaller order terms, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir’s proof of the finite field Kakeya conjecture and the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a lower bound on the size of S dependent on the ideal generated by the homogeneous polynomials vanishing on D. This

bound is maximised as ( 1 2N)n plus smaller order terms, for n 4, when D contains the points of a Nn−1 grid.

bound is maximised as ( 1 2N)n plus smaller order terms, for n 4, when D contains the points of a Nn−1 grid.

Original language | English |
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Pages (from-to) | 251-260 |

Number of pages | 10 |

Journal | The Australasian Journal of Combinatorics |

Volume | 65 |

Publication status | Published - 2016 |