There exists an absolute constant Ξ΄>0 such that for all q and all
subsets AβFqβ of the finite field with q elements, if
β£Aβ£>q2/3βΞ΄, then β£(AβA)(AβA)β£=β£{(aβb)(cβd):a,b,c,dβA}β£>2qβ. Any Ξ΄<1/13,542 suffices for sufficiently large
q. This improves the condition β£Aβ£>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,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.Comment: 42 page