The large sieve through Parseval, Large sifted sets are regular

Source abstract

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that $N\int_{-δ}^δ|S(β)|^2dβ\ge(1-2/\sqrt{δN})|S(0)|^2$ for any trigonometric polynomial of length~$N$.