Theorem 5 — improved large sieve for sifted sequences

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Assume the supporting compact and Johnsen condition. Let $N\geq1$ be an integer and let $\varphi_1,\ldots,\varphi_N$ be complex numbers, extended by zero outside $[1,N]$, supported by $\mathcal K$ up to level $Q$. For $1\leq Q_0\leq Q$,

$$\sum_{q\leq Q_0}\ \sum_{\substack{a\bmod q\\(a,q)=1}} \left|\sum_{n=1}^N\varphi_n e(na/q)\right|^2 \leq \frac{G_1(Q_0)}{G_1(Q/Q_0)}(N+Q^2) \sum_{n=1}^N|\varphi_n|^2.$$

The Fourier sum stops at $Q_0$, but the support condition must hold through $Q$. The denominator is $G_1(Q/Q_0)$, not $G_1(Q)$, and the quadratic cost uses $Q$, not $Q_0$.

The proof on p. 571 compares the nonnegative decomposition in Theorem 4 at the two levels. Lemma 1 bounds the coefficient ratio by $G_1(Q_0)/G_1(Q/Q_0)$; the ordinary large sieve bounds the energy at level $Q$ by $(N+Q^2)\sum_n|\varphi_n|^2$.

For an application to Problem 12(iii), the residue-support system, its uniform lifting, both G-function estimates and the enlarged quadratic cost remain separate requirements. The theorem itself gives no bound for a property-P set's joint divisor exclusions.

Source and reading. Theorem 5 and its proof, printed p. 571 of Ramaré–Ruzsa (2001). The statement and parameter dependence were checked against the scan. The short proof and its cited interfaces were read; no independent proof audit of the large-sieve input or the complete paper is asserted.