An Improved Exponent for Products of Differences over Arbitrary Finite Fields

Murphy and Petridis proved that, for subsets $A$ of an arbitrary finite field $\F_q$, the condition $\abs A>q^{2/3-δ}$ implies $\abs{(A-A)(A-A)}>q/2$ whenever $δ<1/13542$ and $q$ is sufficiently large. We improve the explicit saving to $1/1689$. The proof replaces the main physical-space extraction