Theorem 4 — Fourier-energy identity

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Use the support and G-function conventions. Assume the Johnsen uniform-lifting condition. For a finitely supported complex sequence $(\varphi_n)_{n\geq1}$ supported by $\mathcal K$ up to level $Q\geq1$, put

$$T_\ell(b)=\sum_{n\equiv b\pmod\ell}\varphi_n, \qquad \Theta(q)=|\mathcal K_q|\sum_{b\in\mathcal K_q} \left|\sum_{\ell\mid q}\mu(q/\ell) \frac{|\mathcal K_\ell|}{|\mathcal K_q|}T_\ell(b)\right|^2.$$

Then

$$\sum_{q\leq Q}\ \sum_{\substack{a\bmod q\\(a,q)=1}} \left|\sum_n\varphi_n e(na/q)\right|^2 =\sum_{q\leq Q}G_q(Q)\Theta(q).$$

The numerator in the inner divisor sum is $|\mathcal K_\ell|$, and its denominator is $|\mathcal K_q|$. The sum over $b$ runs over $\mathcal K_q$, whereas the inner residue condition is modulo $\ell$. All positive integer moduli up to $Q$ occur, including prime powers; the sum is not restricted to squarefree moduli. At $q=1$ the reduced residue sum has its single usual term.

Source and reading. Theorem 4, printed p. 569, with proof on p. 570 of Ramaré–Ruzsa (2001). The statement and its support hypotheses were checked against the scan; the proof was read for the residue-fiber and nonnegative-energy interfaces, without an independent proof audit. The theorem assumes uniform lifting, not just multiplicative splitting.