Derivative correlations force additive frequency energy ($E_{\theta }(H)\ge (\beta \eta {)}^4|G{|}^3$)

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.

Use the unnormalized Fourier transform and $|f|\le 1$. Normalize each derivative correlation by $N$ and align its phase. Their absolute sum is at least $\alpha N$. One Cauchy-Schwarz inequality, expansion by shift/frequency differences, and a second Cauchy-Schwarz inequality bound its fourth power by $N$ times the additive energy of a frequency graph. Parseval identity on a finite group supplies the factor $N$; exact equality of both shift sums and frequency sums is retained.