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A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the quantity $E(A):=|\{(a,b,c,d) \in A^4 : a+b=c+d \}|$.

Where in the literature did the term "additive energy" originate?

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    $\begingroup$ Not sure about the term, but the importance of additive energy was realized already by Freiman in mid-sixties. Freiman called it "the invariant $M$''. $\endgroup$ – Seva Nov 18 '15 at 19:38
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Van Vu and I coined the term in our book because there did not seem to be a widely adopted name for it previously. (Gowers, for instance, refers to "number of additive quadruples" rather than "additive energy", but this seemed to be too unwieldy to use for our purposes.) I think we settled on "energy" due to the vaguely quadratic nature of the expression, and also because it was monotone (larger sets have larger energy), in contrast to other statistics measuring additive structure such as the doubling constant. (Coincidentally, in the theory of nonlinear dispersive equations, there are sometimes quartic potential energy terms such as $\frac{1}{4} \int |u|^4$ which can be viewed as essentially being the additive energy of the Fourier transform of the field $u$, so additive energy can in fact be physically interpreted as an energy in some cases, though this was not our intent at the time. Certainly I sometimes visualise an additive quadruple as a kind of "chemical bond" between four "atoms" in a set $A$, so that the additive energy measures something like a "latent energy" in that set.)

I have sometimes thought of referring to the energy $E(A,B) := | \{ (a,b,a',b') \in A \times B \times A \times B: a+b = a'+b' \}$ between two different sets $A$, $B$ as the "synergy" between $A$ and $B$ rather than the "energy", but perhaps it was better that we didn't propose this name as it was probably too clever by half.

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