2012/01/24 by Michael Bennett, Michael A. Bennett, Bennett, Michael +5
Engineering · Mathematics · #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT) #graph theory and CDMA systems #math.CA #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1201.5039
openalex publication_date 2012/01/24 · arxiv created 2012/01/25 · arxiv updated 2012/01/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We prove that if E ⊂ \mathbb Fq2, q ≡ 3 \mod 4, has size greater than Cq7/4, then E determines a positive proportion of all congruence classes of triangles in \mathbb Fq2. The approach in this paper is based on the approach to the Erd\H os distance problem in the plane due to Elekes and Sharir, followed by an incidence bound for points and lines in \mathbb Fq3. We also establish a weak lower bound for a related problem in the sense that any subset E of \mathbb Fq2 of size less than cq4/3 definitely does not contain a positive proportion of \bf translation classes of triangles in the plane. This result is a special case of a result established for n-simplices in \mathbb Fqd. Finally, a necessary and sufficient condition on the lengths of a triangle for it to exist in \mathbbF2 for any field \mathbb F of characteristic not equal to 2 is established as a special case of a result for d-simplices in \mathbb Fd.