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There exists an absolute constant \delta > 0 such that for all q and all subsets AβFqβ 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,XβFqβ for which the number of solutions to the equation (a1ββa2β)=x(a3ββa4β),a1β,a2β,a3β,a4ββA,xβX is nearly maximum. A key ingredient is determining exact algebraic structure of sets A,X for which β£A+XAβ£ 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 (AβB)(CβD)={(aβb)(cβd):aβA,bβB,cβC,dβD} when A,B,C,D are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors
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