Products of Differences over Arbitrary Finite Fields

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.

42 pages

References in corpus (3)

• New results on sum-product type growth over fields
• A sum-product estimate in fields of prime order
• Products of Differences in Prime Order Finite Fields

Cited by in corpus (1)

• On $k$-point configuration sets with nonempty interior