Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that

$$ \sum_{r=1}^R |S(x_r)|^2 \leq (N + \delta^{-1}-1) \int_{0}^{1} |S(x)|^2 dx$$

where $x_1,x_2,\ldots,x_R$ are $\delta$ separated points. This can be thought of as a discretization of Parseval's identity on the circle.

In many of the applications of this inequality to number theory one takes $\delta = 1/N$ in which case the right hand side of the inequality has a factor $2N$.

The factor of $2$ here leads to some very unfortunate inefficiencies in sieve theoretic applications, such as the $2$ in the Brun-Titchmarsh inequality. It seems that improving the factor of $2$ in these applications is related to both the parity problem in sieve theory and the Siegel zero problem. See, for instance, this paper of Maynard.

However it is easy to see in some cases, such as when $x_1,x_2, \ldots, x_R$ are equally spaced (using Fourier analysis on the group of residues mod $N$), that the inequality indeed holds with a constant $1$ in place of the constant $2$.

Are there examples of trigonometric polynomials and $\delta = 1/N$ separated points known where the constant in the analytic large sieve inequality is required to be greater than $1$?

equallyspaced, so it seems that the sentence before your question answers the question. $\endgroup$3more comments