Average estimate for additive energy in prime field

TL;DR

This paper establishes an improved upper bound on the average additive energy between a subset of a prime field and its multiplicative shifts, advancing understanding of additive-multiplicative interactions in finite fields.

Contribution

It provides a new, sharper estimate for the sum of additive energies between a set and its multiplicative shifts in prime fields, improving prior bounds.

Findings

Derived a bound involving p, |A|, |B|, and their ratios.

Showed the bound holds for sufficiently large primes p.

Enhanced previous estimates in additive combinatorics.

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 $p\geqslant p_0(\beta)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{\beta, 1-\alpha\}}{308}}|A|^3|B|.$$ 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.