The first goal of this paper is to prove a sharp condition to guarantee of having a positive proportion of all congruence classes of triangles in given sets in 𝔽 q 2 {\mathbb{F}_{q}^{2}} . More precisely, for A , B , C ⊂ 𝔽 q 2 {A,B,C\subset\mathbb{F}_{q}^{2}} , if | A | ⁢ | B | ⁢ | C | 1 2 ≫ q 4 {|A||B||C|^{\frac{1}{2}}\gg q^{4}} , then for any λ ∈ 𝔽 q ∖ { 0 } {\lambda\in\mathbb{F}_{q}\setminus\{0\}} , the number of congruence classes of triangles with vertices in A × B × C {A\times B\times C} and one side-length λ is at least ≫ q 2 {\gg q^{2}} . In higher dimensions, we obtain similar results for k-simplex but under a slightly stronger condition. Compared to the well-known L 2 {L^{2}} method in the literature, our approach offers better results in both conditions and conclusions. When A = B = C {A=B=C} , the second goal of this paper is to give a new and unified proof of the best current results on the distribution of simplex due to Bennett, Hart, Iosevich, Pakianathan and Rudnev (2017) and McDonald (2020). The third goal of this paper is to study a Furstenberg-type problem associated to a set of rigid motions. The main ingredients in our proofs are incidence bounds between points and rigid motions. While the incidence bounds for large sets are due to the author and Semin Yoo (2023), the bound for small sets will be proved by using a point–line incidence bound in 𝔽 q 3 {\mathbb{F}_{q}^{3}} due to Kollár (2015).

中文翻译：

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具有固定边长的三角形、Furstenberg 型问题以及有限向量空间中的关联

本文的第一个目标是证明一个锐条件来保证给定集合中三角形的所有同余类具有正比例 𝔽 q 2 {\mathbb{F}_{q}^{2}} 。更准确地说，对于 A , 乙 , C ⊂ 𝔽 q 2 {A,B,C\子集\mathbb{F}_{q}^{2}} ， 如果 | A | ⁢ | 乙 | ⁢ | C | 1 2 ≫ q 4 {|A||B||C|^{\frac{1}{2}}\gg q^{4}} ，那么对于任意 λ ε 𝔽 q ∖ { 0 } {\lambda\in\mathbb{F}_{q}\setminus\{0\}} ，顶点为 的三角形的同余类数量 A × 乙 × C {A\乘以B\乘以C} 并且一条边长 λ 至少为 ≫ q 2 {\gg q^{2}} 。在更高的维度中，我们获得了类似的结果k-单纯形但条件稍强。与众所周知的相比 L 2 {L^{2}} 与文献中的方法相比，我们的方法在条件和结论方面都提供了更好的结果。什么时候 A = 乙 = C {A=B=C} ，本文的第二个目标是对 Bennett、Hart、Iosevich、Pakianathan 和 Rudnev (2017) 以及 McDonald (2020) 提出的单纯形分布的当前最佳结果给出新的统一证明。本文的第三个目标是研究与一组刚性运动相关的弗斯滕伯格型问题。我们证明中的主要成分是点和刚性运动之间的重合边界。虽然大集合的关联界限是由作者和 Semin Yoo (2023) 提出的，但小集合的界限将通过使用点线关联界限来证明 𝔽 q 3 {\mathbb{F}_{q}^{3}} 感谢 Kollár (2015)。