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arXiv · 2003.03493

Bounds of Trilinear and Trinomial Exponential Sums

Simon Macourt·Giorgis Petridis·Ilya D. Shkredov·Igor E. Shparlinski
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Abstract

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.

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Simon Macourt, Giorgis Petridis, Ilya D. Shkredov, Igor E. Shparlinski. 2020-03-07. Bounds of Trilinear and Trinomial Exponential Sums. https://doi.org/10.1137/20m1325502

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