Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = |\lbrace(a_1,a_2,a_3,a_4) \in A^4 : a_1 + a_2 = a_3 + a_4 \rbrace|$.

A related quantity that I'm interested in is: $F(A) = |\lbrace (a_1,a_2) \in A^2 : a_1 + a_2 \in A \rbrace|$. It seems to me that $F(A)$ captures the notion of "closed under sumset" more directly. How come $F(A)$ isn't studied more in additive combinatorics? What kinds of statements can one make about the relationship between $F(A)$ and $E(A)$?

Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = |\lbrace(a_1,a_2,a_3,a_4) \in A^4 : a_1 + a_2 = a_3 + a_4 \rbrace|$.

A related quantity that I'm interested in is: $F(A) = |\lbrace (a_1,a_2) \in A^2 : a_1 + a_2 \in A \rbrace|$. It seems to me that $F(A)$ captures the notion of "closed under sumset" more directly. How come $F(A)$ isn't studied more in additive combinatorics? What kinds of statements can one make about the relationship between $F(A)$ and $E(A)$?

In particular, I'm mostly concerned with situations when $G$ is a vector space like $\mathbb{F}^n$ for some finite field $\mathbb{F}$.