Average estimate for additive energy in prime field

Abstract

Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^{\alpha },|B|=p^{\beta }$. We will prove that for $p⩾p_0(\beta )$ one has $$\sum _{b\in B}{E}_{+}(A,bA)⩽15p^{-\frac{\min \{\beta ,1-\alpha \}}{308}}|A{|}^3|B|.$$ Here $E_{+}(A,bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.

arXiv comment19 pages