AREAS OF TRIANGLES AND BECK'S THEOREM IN PLANES OVER FINITE FIELDS

The first main result of this paper establishes that any sufficiently large subset of a plane over the finite field F-q, namely any set E subset of F-q(2) of cardinality vertical bar E vertical bar > q, determines at least 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 vertical bar E vertical bar = Omega(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 is an element of 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 F-q(2). Namely, if vertical bar E vertical bar = Omega(q log q), then a positive proportion of points z is an element of E has a property that there are Omega(q) straight lines incident to z, each supporting, up to constant factors, approximately the expected number vertical bar E vertical bar/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 is an element of E; unless vertical bar E vertical bar = Omega(q(3/2)). We also briefly discuss higher-dimensional implications of these results in light of some recent developments in the literature.