2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ that Fourier series looks completely arbitrary. or just a trigonometric series since it's finite. So this large Sieve just says this trig average is bounded by some average of the coefficients? This also looks like a contractive inequality between two $L^2$ spaces. $\endgroup$ – john mangual Feb 15 '17 at 15:29
  • $\begingroup$ In addition to the series of papers by Huxley, you should also read Gallagher's paper The large sieve and probabilistic Galois theory. Regarding other generalizations, there is also (of course) a variant of this for $\mathbb{F}_q[T]$. $\endgroup$ – Vesselin Dimitrov Feb 15 '17 at 16:07
5
$\begingroup$

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 $$\prod_{i=1}^{n} (N_i^{1/2} + \delta_i^{-1/2})^2.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.