= Solution
Use the same <Sobolev–Gallagher inequality> on arcs of length $\Delta$. They now overlap, but every point belongs to at most $K(\Delta)$ arcs, by the definition of that local multiplicity. For $0<\Delta\leq1$, summing the integrals therefore gives
$$
\sum_r|S(\theta_r)|^2\leq K(\Delta)\left(\Delta^{-1}\int_0^1|F|^2+\int_0^1|FF\prime|\right)
\leq K(\Delta)(\Delta^{-1}+2\pi N)E.
$$
This is the <local-multiplicity large sieve>. If $\Delta\geq1$, every point is within circular distance $\Delta/2$ of every center, so $K(\Delta)=R$. The direct <Cauchy-Schwarz inequality> bound $\sum_r|S(\theta_r)|^2\leq RNE$ gives the requested estimate in this remaining case.
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