# Products of Differences over Arbitrary Finite Fields

Brendan Murphy, Giorgis Petridis

Research output: Contribution to journalArticle (Academic Journal)peer-review

## Abstract

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then $|(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| > \frac{q}{2}.$ Any $\delta <1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on a qualitatively optimal characterisation of sets $A,X \subseteq \mathbb{F}_q$ for which the number of solutions to the equation $(a_1-a_2) = x (a_3-a_4) \, , \; a_1,a_2, a_3, a_4 \in A, x \in X$ is nearly maximum. A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. We also prove a stronger statement for $(A-B)(C-D) = \{ (a -b) (c-d) : a \in A, b \in B, c \in C, d \in D\}$ when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.
Original language English Discrete Analysis Submitted - 18 May 2017

45 pages

• math.CO
• math.NT
• 11B30