SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2124-2136, January 2020.
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of elements of $\mathcal{A}$. We also prove a refined estimate on the number of collinear triples in a Cartesian product of multiplicative subgroups and derive stronger bounds for trilinear sums with all variables in multiplicative subgroups.

中文翻译：

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三线性和三项指数和的界

SIAM离散数学杂志，第34卷，第4期，第2124-2136页，2020年1月。
我们证明，对于素数残基字段的足够小的子集$ \ mathcal {A} $，可以估算出该问题的解数。等式$（a_1-a_2）（a_3-a_4）=（a_5-a_6）（a_7-a_8）$，所有变量都在$ \ mathcal {A} $中。然后，我们得出三线性指数和以及等于$ \ mathcal {A} $元素的两个差的乘积的残数总数的新界限。我们还证明了对乘法子群的笛卡尔积中共线三元组数的精确估计，并为乘法子群中所有变量的三线性和得出了更强的边界。