Return to Answer

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 (Bulletin of the AMS, 1978) 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 $x_j$'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 $x_j$'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, $$\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2=\sum_{j,k}c_j\overline{c_k}\sum_{n=1}^N e^{2\pi in(x_j-x_k)}.$$ The inner sum equals $N$ for $j=k$, and it is bounded by $|\sin\pi(x_j-x_k)|^{-1}$ in absolute value for $j\neq k$. The result follows.

In fact the above argument coupled with Corollary 1 in Montgomery: The analytic principle of the large sieve (Bulletin of the AMS, 1978) 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 $x_j$'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 (Bulletin of the AMS, 1978) 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 $x_j$'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, $$\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2=\sum_{j,k}c_j\overline{c_k}\sum_{n=1}^N e^{2\pi in(x_j-x_k)}.$$ The inner sum equals $N$ for $j=k$, and it is bounded by $|\sin\pi(x_j-x_k)|^{-1}$ in absolute value for $j\neq k$. The result follows.

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, $$\sum_{n=1}^{N}\left|\sum_{j}c_j e^{2\pi i n x_j}\right|^2=\sum_{j,k}c_j\overline{c_k}\sum_{n=1}^N e^{2\pi in(x_j-x_k)}.$$ The inner sum equals $N$ for $j=k$, and it is bounded by $|\sin\pi(x_j-x_k)|^{-1}$ in absolute value for $j\neq k$. The result follows.

In fact the above argument coupled with Corollary 1 in Montgomery: The analytic principle of the large sieve (Bulletin of the AMS, 1978) 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 $|\Delta|\leq\max_{j\neq k}\|x_j-x_k\|^{-1}.$