Products of Differences over Arbitrary Finite Fields

There exists an absolute constant δ > 0 such that for all q and all subsets A ⊆ Fq of the finite field with q elements, if |A| > q2/3−δ , then |(A−A)(A−A)| = |{(a− b)(c − d) : a, b, c, d ∈ A}| > q 2 . Any δ < 1/13, 542 suffices for sufficiently large q. This improves the condition |A| > q2/3, 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 ⊆ Fq for which the number of solutions to the equation (a1 − a2) = x(a3 − a4) , a1, a2, a3, a4 ∈ A, x ∈ 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 ∈ A, b ∈ B, c ∈ C, d ∈ D} when A,B,C,D are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.