A conjecture about generalized additive energy

Question context

Let $A\subset G$ and define $E_{2n}(A) = |\{(a_1,\cdots,a_n)\in A^{2n}: a_1+\cdots+a_n = a_{n+1}+\cdots+a_{2n}\}|.$ Trivial bounds on additive energy are $|A|^n\leq E_{2n}(A)\leq |A|^{2n-1}.$ Define $\rho_{2n}(A)= \frac{E_{2n}(A)}{|A|^n}.$

Given this, one can compare various $\rho_k$'s. For example by an applications of Holder's inequality one can get

Moreover, there is the following theorem.

Theorem. Fix some $0<\varepsilon<1$. Suppose $\rho_8(A)\leq K\rho_4^3(A)$. then there is a subset $H\subset A$, so that

$|H|\geq \rho_4(A)/K^c$ and $E_4(H)\geq \frac{|H|^3}{|A|^{\varepsilon}K^{c'}}.$

Now, using Holder's inequality one can show

Historical workshop record. Current open/solved status has not been independently reassessed.

Status note in the source

This conjecture has applications to PFR. In fact, given a set $A$ , we can list all the energies for various $n$, and separate to two cases.

1. Energy always increases (Smoothing.)

2. There is one point that the assumption of the conjecture is satisfied.(Non-smoothing) . The conjecture will give us some structure.