Exponential-sum large sieve

If ${\theta }_r$ have circular spacing at least $\delta$, thenMultiply 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 $2\pi N\sum |a_n{|}^2$.

Let $K(\Delta )={\max }_t\#\{r:\Vert {\theta }_r-t\Vert \le \Delta /2\}$. Without a separation assumption,For $0<\Delta \le 1$, each point belongs to at most $K(\Delta )$ integration arcs in the Sobolev–Gallagher inequality. For $\Delta \ge 1$, $K(\Delta )$ is the total number of points, and the Cauchy-Schwarz inequality bound $|S{|}^2\le N\sum |a_n{|}^2$ suffices.

When $N\le P$,An arc of length $1/P$ contains at most three points of each grid of denominator $p\le 2P$. The Chebyshev estimate gives $O(P/\log P)$ such primes. Apply the local-multiplicity large sieve. Using only the separation of distinct reduced fractions gives the weaker $O(P^2)\sum |a_n{|}^2$.