A measure of closure under sumset?

Question

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 $\mathbb{F}$.

Answer 1

On the one hand there are some notions that seem related that are studied (see at the end), but on the other hand the precise defintion you give does not have some, at least from a cetain point of view, desirable features.

First, on this second aspect an example (there are various other 'good' properties of the additive energy but I focus on this one only, here).

An arithmetic progression is considered as a very "additively closed" set, in some sense it is "as closed as possible"; for example in the sense that (finite) arithmetic progressions are precisley the sets, in the integers (or more generally torsion free abelian groups), $P$ such that $|P+P|=2 |P| -1$ and they are optimal at this as $|A+A| \ge 2 |A| -1$ for any finite set.

Now, your $F(A)$ is indeed small for some arithmetic progressions (indeed, I think, as small as possible when fixing the cardinality of the set) but huge (as large as possible) for others, (for certain applications) this is not desirable; as an example compare the results for initial segments of the even and odd natural numbers, respectively. While the additive energy would always be the same for arithemtic progressions, of a given lengths. So I do not think there is much direct relation here.

To put this differently it is desirable that such notions are rather invariant under affine transformations; in particular, $F(A) = F(b+A)$ would be something that is nice to have. The additive energy has this property, your notion does not.

But, having said that there are investigations in Additive Combinatorics that do go in the direction you seem to have in mind. Specifically, there is the Isoperimetric Method, due to the late Hamidoune, that very roughly speaking is based on considering how much a 'shift' of a set (by certain elements) will differ from the original set, or how much sets 'expand' in certain Cayley graphs.

See for example one of his papers "Topology of Cayley Graphs Applied to Inverse Additive Problems" (but there are many more due to him and others based on this approach; look for example for his papers on arXiv). Note in particular the definition of the boundary of a set there; while this does not match exactly what you envision I thought it is somewhat close in spirit and you thus might find it interesting, and there might be things closer still but unfortunately I do not have a very good overview.

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Added after restriction to finite fields setting in the question:

In this context the 'problem' mentioned above persists. Here, or then also in full generality (any abelian group), besides arithmetic progression, sets close to additively closed are cosets (that is affine subspaces, in the vectorspace setting) and the mix of the two that is 'progressions of cosets'. And, your notion does not behave well here either, in the sense of being small when a set is (close to) such a set, but sometimes it is small sometimes it is large.

More generally, beyond 'affine transformation' it is good if such notions behave well under Freiman isomorphisms. The additive energy is invariant under Freiman isomorphisms (of order two) while your notion is not. (A map of the form $a+f(⋅)$ with $f(⋅)$ an injective group homomorphism is an example of a Freiman isomorphism.)

Or put differently, several of the classical results of the subject, e.g., Theorems of Cauchy-Davenport, Chowla, Vosper, Kneser, Freiman, are concerenced with considering $|A+ A|$ (relative to $|A|$, or $|A +B|$ relative to $|A|+|B|$ and so on), not so much the relation of $A+A$ and $A$ itself. So one uses/wants notions well-suited to study these types of problems.

Answer 2

The energy parameter is translation invariant, meaning that

$$E(A+x) = E(A)$$

where   $A+x\ :=\ \{a+x:a\in A\}$.   Your force parameter does not enjoy this kind of invariance.

For many important subsets   $A\subset G$   your force will be   $0$   thus providing no useful distinction between them.

But then, who knows, there might be situations which your force fits very well. For instance, when we don't know whether or not   $0\in A$   while  $F(A)$   has a high value. Then it is likely that   $0\in A$.

REMARK   One can say that energy is even affine since it is also linear, meaning that:

$$E(T(A)) = E(A)$$

for every monomorphism   $T:A\rightarrow A$.   Of course your force is linear too (but not affine).