Areas of triangles and Beck’s theorem in planes over finite fields

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 qo(q) distinct areas of triangles sharing a common vertex z, this property shared by a positive proportion of zE. 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 zE 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 zE; 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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Correspondence to Misha Rudnev.

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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