One of the most important achievements in analytic number theory is the establishment of the so-called large sieve inequality, which is formulated as follows. Let $\{a\_n\}$ denote a finite sequence of complex numbers, say supported on the segment $M \leq n < M + N$. For a real number $\alpha$ put


$$\displaystyle S(\alpha) = \sum\_n a\_n e(\alpha n),$$


where $e(x) = \exp(2\pi i x)$. Let $\alpha\_1, \cdots, \alpha\_r$ be real numbers such that $\lVert \alpha\_i - \alpha\_j \rVert \geq \delta > 0$ for $i \ne j$. Here $\lVert x \rVert$ of a real number $x$ denotes the distance of $x$ to the nearest integer. The large sieve inequality (of Selberg and Montgomery-Vaughan, independently) then asserts that for any finite sequence of complex numbers $\{a\_n\}$ supported on $N$ integers we have


$$\displaystyle \sum\_j |S(\alpha\_j)|^2 \leq (\delta^{-1} + N - 1) \sum\_n |a\_n|^2$$


and that this inequality is best possible in general. 


One can formulate a higher dimensional analogue of this. Let $\mathbf{a} = (a\_1, \cdots, a\_n)$ be a vector of real numbers, and let $G\_{\mathbf{v}}$ be complex numbers supported on the box $B = \{\mathbf{v} \in \mathbb{Z}^n : M \leq v\_i < M + N\}$. Put


$$\displaystyle S(\mathbf{a}) = \sum\_{\mathbf{v} \in B} G\_{\mathbf{v}} \exp(2 \pi i \mathbf{a} \cdot \mathbf{v}).$$


Suppose that $\mathbf{a}\_1, \cdots, \mathbf{a}\_r$ are vectors which are pairwise separated by $\delta$ modulo 1. Is there a general formula for good functions $F\_n(\delta, N)$ for which the inequality


$$\displaystyle \sum\_j |S(\mathbf{a}\_j)|^2 \leq F\_n(\delta, N) \sum\_{\mathbf{v} \in B} |G\_{\mathbf{v}}|^2?$$


For instance, we can take $F\_1(\delta, N) = \delta^{-1} + N - 1$. 



Higher dimensional large sieve inequality
There is a paper of Huxley from 1968 (<http://www.ams.org/mathscinet-getitem?mr=237455>) that answers this. If I am translating the notation correctly, Huxley shows that
$$F\_n(\delta, N) = (N^{1/2} + \delta^{-1/2})^{2n}$$
is allowable. More generally, if one restricts to $M\_i \leq v\_i < M\_i + N\_i$ and assumes that the $i$-th component of ${\bf a}\_k - {\bf a}\_{\ell}$ is spaced by at least $\delta\_i$ (modulo $1$, $k \neq \ell$), then one can replace $F\_n(\delta, N)$ by $$\prod\_{i=1}^{n} (N\_i^{1/2} + \delta\_i^{-1/2})^2.$$


