Are there general large sieve lower bounds?

Question

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, 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.

Answer 1

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 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 $|\Delta|\leq\max_{j\neq k}\|x_j-x_k\|^{-1}.$