Sets Avoiding Full-Rank Three-Point Patterns in (\mathbbF_qⁿ)^k Are Exponentially Small

2022/08/30 by Mohamed Omar, Omar, Mohamed
Mathematics · #05D40 #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2208.14266

openalex publication_date 2022/08/30 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

We prove that if a subset of (\mathbbF_qⁿ)^k (with q an odd prime power) avoids a full-rank three-point pattern x,x+M₁d,x+M₂d then it is exponentially small, having size at most 3 ⋅ c_q^nk where 0.8414 q ≤ c_q ≤ 0.9184 q. This generalizes a theorem of Kovauc and complements results of Berger, Sah, Sawhney and Tidor. As a consequence, we prove that if 3 is a square in \mathbbF_q then subsets of (\mathbbF_qⁿ)² avoiding equilateral triangles are exponentially small.

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