We let \(\alpha_{1},...,\alpha_{R}\) be \(R\) real numbers that are pairwise distinct modulo one and we let \(a=(a_{n})_{M<n\leq M+N}\) be an arbitrary \(N\)-vector of complex numbers. We let further

\[S(\alpha)=\sum_{M<n\leq M+N}a_{n}e(\alpha n),\]

where as usual \(e(x):=\exp (2\pi ix)\).

Then

\[\sum_{r\leq R}|S(\alpha)|^{2}\leq(N-1+\delta^{-1})\left\Vert a\right\Vert ^{2},\]

where \(\delta\) is the minimum distance between different \(\alpha_{i}\) in \(\mathbb{R}/\mathbb{Z}\). The above is an example of a “Large Sieve”-inequality …