An Improved Exponent for Products of Differences over Arbitrary Finite Fields

Abstract

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 $\delta <1/13542$ and $q$ is sufficiently large. We improve the explicit saving to $1/1689$. The proof replaces the main physical-space extraction in their large-energy branch by a common Fourier spectrum, followed by a popular-ratio Balog--Szemerédi--Gowers step. The resulting additive energy is treated level by level, retaining difference multiplicities and combining the Reiher--Schoen relation lemma with Mohammadi's difference--ratio estimate. This gives the structural exponent $188$ and hence \[ \abs A\ge Cq^{375/563}=Cq^{2/3-1/1689} \quad\Longrightarrow\quad \abs{(A-A)(A-A)}>\frac q2 \] for an absolute constant $C>0$.

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