Large energy implies many popular pairs

Let $G$ be an additive commutative group and $X, Y \subseteq G$ finite sets with $X$ nonempty and $|X| = |Y|$. Suppose

where $E(X,Y) = \sum_s r_{X,Y}(s)^2$ is the additive energy and $r_{X,Y}(s) = X.\mathrm{addConvolution}\, Y\, s$ counts representations $s = x+y$. If $\theta : \mathbb{N}$ satisfies $2\theta \le \eta\,|X|$, then the popular pairs are numerous:

The proof splits the energy sum into popular and unpopular fibres and uses the total-mass identity $\sum_s r(s) = |X||Y|$. This lemma quantifies the first half of the energy-to-graph conversion: truncating the convolution at level $\theta$ retains a constant fraction of all pairs, so that together with the Markov upper bound on the number of popular sums it produces the dense popular-sum graph.