Exercise 57, § 1

On retains the notations of the preceding exercise. Let M, $N\geqslant 1$ be integers and let $(a_n)_{M<n\leqslant M+N}$ be a family of complex numbers. Let $\delta >0$ be a real number, $R\geqslant 1$ an integer and $(\xi_r)_{1\leqslant r\leqslant R}$ a family of real numbers such that inf$_{h\in\mathbf{Z}}|\xi_r-\xi_{r'}-h|\geqslant \delta$ for all $r\not=r'$.

For every $x\in \mathbf{R}$, put

a) One has

("large sieve inequality").

(Consider the function $\tau :z\mapsto \sigma (\delta z)$, where $\sigma$ is the function of questions f) and g) of the preceding exercise for the interval $[\alpha , \beta ] = [\delta (M + 1), \delta (M + N)]$; put

and observe that

Put $x=\xi_r$, apply the Cauchy-Schwarz inequality and sum over $r$.)

b) The constant $N +\delta^{-1}-1$ is optimal.