Given a set of distinct real numbers $(x\_j)$ and non-zero complex $(c\_j)$, then the large sieve says that


$$\limsup\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2\leq \sum\_{j}|c\_j|^2.$$


Are there lower bounds saying that


$$\liminf\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2\geq \varepsilon \,\,?$$


I am aware of the paper titled [Lower bounds for expressions of large sieve type](https://arxiv.org/pdf/1105.1307.pdf), but this deals with a different formulation than the one I am interested in. Namely, it only refers to the dual equality and it includes more terms than I do. Any help would be greatly appreciated.



Are there general large sieve lower bounds?
If the $x\_j$'s are distinct modulo $1$ (which is the natural assumption), then
$$\lim\_{N\to\infty}\frac{1}{N}\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2=\sum\_{j}|c\_j|^2.$$
Indeed, let us assume (without loss of generality) that the $x\_j$'s lie in $[0,1]$. Then
$$\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2=\sum\_{j,k}c\_j\overline{c\_k}\sum\_{n=1}^N e^{2\pi in(x\_j-x\_k)},$$
where the inner sum equals $N$ for $j=k$, and has absolute value not exceeding $\csc(\pi(x\_j-x\_k))$ for $j\neq k$. The result follows.


**Remark.** The above argument coupled with Corollary 1 in [Montgomery: The analytic principle of the large sieve](https://projecteuclid.org/euclid.bams/1183540922) gives that
$$\sum\_{n=1}^{N}\left|\sum\_{j}c\_j e^{2\pi i n x\_j}\right|^2=(N+\Delta)\sum\_j|c\_j|^2,$$
where $|\Delta|\leq\max\_{j\neq k}\|x\_j-x\_k\|^{-1}.$


