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 $F_1(\delta, N) = \delta^{-1} + N - 1$.

contractiveinequality between two $L^2$ spaces. $\endgroup$The large sieve and probabilistic Galois theory. Regarding other generalizations, there is also (of course) a variant of this for $\mathbb{F}_q[T]$. $\endgroup$