 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 $0<\Delta \le 1$, summing the integrals therefore givesThis is the local-multiplicity large sieve. If $\Delta \ge 1$, every point is within circular distance $\Delta /2$ of every center, so $K(\Delta )=R$. The direct Cauchy-Schwarz inequality bound ${\sum }_r|S({\theta }_r){|}^2\le RNE$ gives the requested estimate in this remaining case.