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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.

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    $\begingroup$ Look in the literature on Turan's power sum method. For example there is a chapter on this in Montgomery's Ten Lectures book. $\endgroup$
    – Lucia
    Commented Jan 17, 2022 at 19:18

1 Answer 1

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

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    $\begingroup$ if the OP wants something more precise for the error term, it might make sense to put in the Fejar kernel and use $$\sum_{n=1}^{N}\left(1-\frac{n}{N+1}\right)\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2.$$ $\endgroup$ Commented Jan 18, 2022 at 0:21
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    $\begingroup$ @JoeSilverman See my added "Remark". $\endgroup$
    – GH from MO
    Commented Jan 18, 2022 at 0:44

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