Tools on Fourier transforms
===========================

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1. The large sieve inequality
-----------------------------


The best version of the large sieve inequality from
:cite:`Montgomery-Vaughan74`
and
:cite:`Montgomery-Vaughan73`
(obtained at the same time by A. Selberg) is as follows.

 
.. admonition:: Theorem (1974)
   :class: thm-tme-emt

   Let :math:`M` and :math:`N\ge 1` be two real numbers. Let :math:`X` be a set of points of
   :math:`[0,1)` such that 
   :math:`\displaystyle\min_{\substack{x,y\in X\\ x\neq y}}\min_{k\in\mathbb{Z}}|x-y+k|\ge \delta>0.`
   Then, for any sequence of complex numbers :math:`(a_n)_{M < n\le M+N}`, we have
   :math:`\displaystyle  \sum_{x\in X}\left|\sum_{M < n\le M+N} a_n \exp(2i\pi nx)\right|^2\le \sum_{M < n\le M+N}|a_n|^2 (N-1+\delta^{-1}).` 



It is very often used with part of the Farey dissection.

 
.. admonition:: Theorem (1974)
   :class: thm-tme-emt

   Let :math:`M` and :math:`N\ge 1` be two real numbers. Let :math:`Q\ge1` be a real parameter.
   For any sequence of complex numbers :math:`(a_n)_{M < n\le M+N}`, we have
   :math:`\displaystyle  \sum_{q\in Q}\sum_{\substack{a\mod q,\\ (a,q)=1}}\left|\sum_{M < n\le M+N} a_n \exp(2i\pi na/q)\right|^2\le \sum_{M < n\le M+N}|a_n|^2 (N-1+Q^2).` 


The summation over :math:`a` runs over all invertible (also called reduced)
classes :math:`a` modulo :math:`q`.

The weighted large sieve inequality from Theorem 1 in
:cite:`Montgomery-Vaughan74`
reads as follows.

 
.. admonition:: Theorem (1974)
   :class: thm-tme-emt

   Let :math:`M` and :math:`N\ge 1` be two real numbers. Let :math:`X` be a set of points of
   :math:`[0,1)`. Define
   :math:`\displaystyle  \delta(x)=\min_{\substack{y\in X\\ y\neq x}}\min_{k\in\mathbb{Z}}|x-y+k|.` 
   Then, for any sequence of complex numbers :math:`(a_n)_{M < n\le M+N}`, we have
   :math:`\displaystyle  \sum_{x\in X}\bigl(N+c\delta(x)^{-1}\bigr)^{-1}\left|\sum_{M < n\le M+N} a_n \exp(2i\pi nx)\right|^2\le \sum_{M < n\le M+N}|a_n|^2` 	      for :math:`c=3/2`.


It is expected that one can reduce the constant :math:`c=3/2` to 1. In this
direction, we find in
:cite:`Preissmann84`
the next result.

 
.. admonition:: Theorem (1984)
   :class: thm-tme-emt

   We may take :math:`\displaystyle c= \sqrt{1+\tfrac23\sqrt{\tfrac65}} =1.3154\cdots < 4/3` in the previous theorem.



See
:cite:`Yangjit23`
for a discussion on this topic.