## Abstract

The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field \(\mathbb{F}_q\), namely any set \(E \subseteq \mathbb{F}_q^2\) of cardinality |*E*| > *q*, determines at least \(\tfrac{{q - 1}} {2}\) 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 \(\mathbb{F}_q^2\). 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 \(\tfrac{{\left| E \right|}} {q}\) 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 \(\left| E \right| = \Omega \left( {q^{\tfrac{3} {2}} } \right)\).

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

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The authors were partially supported by the NSF Grant DMS-1045404

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Iosevich, A., Rudnev, M. & Zhai, Y. Areas of triangles and Beck’s theorem in planes over finite fields.
*Combinatorica* **35, **295–308 (2015). https://doi.org/10.1007/s00493-014-2977-7

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### Mathematics Subject Classification (2000)

- 68R05
- 11B75