Additive energy of a frequency graph ($E(\{(h,\theta (h)):h\in H\})$)

For a set of shifts $H$ and a map $\theta$ into a circle group, this additive energy counts quadruples whose shift sums and frequency sums both agree. It is the additive energy of the graph of $\theta$ in the product abelian group.

Suppose $|f|\le 1$ on a finite cyclic group, $H$ has density $\beta$, and for each $h\in H$ the multiplicative derivative has a Fourier coefficient on a finite abelian group of magnitude at least $\eta$ at $\theta (h)$. Align the correlation phases, apply Cauchy-Schwarz, group pairs by their shift and frequency differences, and apply Parseval identity. This proves the displayed lower bound on additive energy of a frequency graph without replacing exact frequency equalities by approximate ones.