An improvement on the number of simplices in \mathbbF_q^d

2016/08/23 by Thang Pham, Pham, Thang, Duc Hiep Pham +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.1608.06398

openalex publication_date 2016/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E be a set of points in \mathbbF_q^d. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if |E|≫ q^{d-(d-1)/(k+1)} then E determines a positive proportion of all k-simplices. In this paper, we give an improvement of this result in the case when E is the Cartesian product of sets. More precisely, we show that if E is the Cartesian product of sets and q^{(kd)/(k+1-1/d)}=o(|E|), the number of congruence classes of k-simplices determined by E is at least (1-o(1))q^\binomk+12, and in some cases our result is sharp.

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