Bounds of Trilinear and Trinomial Exponential Sums

Simon Macourt; Giorgis Petridis; Ilya D. Shkredov; Igor E. Shparlinski

doi:10.1137/20M1325502

paper

Bounds of Trilinear and Trinomial Exponential Sums

arXiv:2003.03493 · doi:10.1137/20M1325502

Abstract

We prove, for a sufficiently small subset $A$ 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 $A$ . 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 $A$ . 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.

15 pages