论文标题
集合避免在$（\ mathbb {f} _q^n）^k $的全级三点模式中
Sets Avoiding Full-Rank Three-Point Patterns in $(\mathbb{F}_q^n)^k$ Are Exponentially Small
论文作者
论文摘要
我们证明，如果$（\ Mathbb {f} _q^n）^k $的子集（$ q $ a od prime功率）避免了完整的三分式图案$ \ vec {x}，\ vec {x}}+m_1+m_1 \ m_1 \ m_1 \ vec {d}最多$ 3 \ cdot c_q^{nk} $其中$ 0.8414 q \ leq c_q \ leq 0.9184 q $。这概括了科沃克的定理，并补充了伯格，萨希，索赫尼和潮汐的结果。结果，我们证明，如果$ 3 $是$ \ mathbb {f} _q $中的一个平方，则$（\ mathbb {f} _q^n）^2 $避免平衡三角形的子集很小。
We prove that if a subset of $(\mathbb{F}_q^n)^k$ (with $q$ an odd prime power) avoids a full-rank three-point pattern $\vec{x},\vec{x}+M_1\vec{d},\vec{x}+M_2\vec{d}$ then it is exponentially small, having size at most $3 \cdot c_q^{nk}$ where $0.8414 q \leq c_q \leq 0.9184 q$. 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 $\mathbb{F}_q$ then subsets of $(\mathbb{F}_q^n)^2$ avoiding equilateral triangles are exponentially small.
