## Abstract

The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field FqFq, namely any set E⊆F2qE⊆Fq2 of cardinality |*E*| > *q*, determines at least q−12q−12 distinct areas of triangles. Moreover, one can find such triangles sharing a common base in *E*, and hence a common vertex. However, we stop short of being able to tell how “typical” an element of *E* such a vertex may be.

It is also shown that, under a more stringent condition |*E*| = *Ω*(*q* log *q*), there are at least *q* − *o*(*q*) distinct areas of triangles sharing a common vertex *z*, this property shared by a positive proportion of *z* ∈ *E*. This comes as an application of the second main result of the paper, which is a finite field version of the Beck theorem for large subsets of F2qFq2. Namely, if |E| = *Ω*(*q* log *q*), then a positive proportion of points *z* ∈ *E* has a property that there are *Ω*(*q*) straight lines incident to *z*, each supporting, up to constant factors, approximately the expected number |E|q|E|q of points of *E*, other than *z*. This is proved by combining combinatorial and Fourier analytic techniques. A counterexample in [14] shows that this cannot be true for every *z* ∈ *E*; unless |E|=Ω(q32)|E|=Ω(q32).

We also briefly discuss higher-dimensional implications of these results in light of some recent developments in the literature.

Original language | Undefined/Unknown |
---|---|

Pages (from-to) | 295-308 |

Number of pages | 14 |

Journal | Combinatorica |

Volume | 35 |

Issue number | 3 |

Early online date | 22 Aug 2014 |

DOIs | |

Publication status | Published - 1 May 2015 |

## Keywords

- math.CO
- math.CA
- math.NT
- 52C10