Exponential-sum large sieve
= Exponential-sum large sieve
If $\theta_r$ have <circular spacing> at least $\delta$, then
$$
\sum_r\left|\sum_{M<n\leq M+N}a_ne(n\theta_r)\right|^2\leq(\delta^{-1}+2\pi N)\sum_{M<n\leq M+N}|a_n|^2.
$$
Multiply the <exponential sum> by $e(-Mt)$, apply the <Sobolev–Gallagher inequality> on disjoint arcs of length $\delta$, and sum. The <finite-interval Parseval identities> and <Cauchy-Schwarz inequality> bound the derivative contribution by $2\pi N\sum|a_n|^2$.