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

2012/05/01 by Alex Iosevich, Iosevich, Alex, Misha Rudnev +3
Mathematics · #52C10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.CO #math.NT #msc:52C10

paper · pdf · doi:10.48550/arxiv.1205.0107

arxiv created 2012/05/01 · arxiv updated 2012/05/02

Abstract

It is shown that any subset E of a plane over a finite field \F_q, of cardinality |E|>q determines not less than (q-1)/(2) distinct areas of triangles, moreover once can find such triangles sharing a common base. It is also shown that if |E|≥ 64qlog₂ q, then there are more than (q)/(2) distinct areas of triangles sharing a common vertex. The result follows from a finite field version of the Beck theorem for large subsets of \F_q² that we prove. If |E|≥ 64qlog₂ q, there exists a point z∈ E, such that there are at least (q)/(4) straight lines incident to z, each supporting the number of points of E other than z in the interval between (|E|)/(2q) and (2|E|)/(q). This is proved by combining combinatorial and Fourier analytic techniques. We also discuss higher-dimensional implications of these results in light of recent developments.

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