# Are there general large sieve lower bounds?

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.

• Look in the literature on Turan's power sum method. For example there is a chapter on this in Montgomery's Ten Lectures book. Jan 17, 2022 at 19:18

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 $$|\Delta|\leq\max_{j\neq k}\|x_j-x_k\|^{-1}.$$
• if the OP wants something more precise for the error term, it might make sense to put in the Fejar kernel and use $$\sum_{n=1}^{N}\left(1-\frac{n}{N+1}\right)\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2.$$ Jan 18, 2022 at 0:21