Local-multiplicity large sieve
= Local-multiplicity large sieve
{title2=$K(\Delta)$}
Let $K(\Delta)=\max_t\#\{r:\|\theta_r-t\|\leq\Delta/2\}$. Without a separation assumption,
$$
\sum_r|S(\theta_r)|^2\leq K(\Delta)(\Delta^{-1}+2\pi N)\sum|a_n|^2.
$$
For $0<\Delta\leq1$, each point belongs to at most $K(\Delta)$ integration arcs in the <Sobolev–Gallagher inequality>. For $\Delta\geq1$, $K(\Delta)$ is the total number of points, and the <Cauchy-Schwarz inequality> bound $|S|^2\leq N\sum|a_n|^2$ suffices.