Tools on Fourier transforms

1. The large sieve inequality

The best version of the large sieve inequality from [Montgomery and Vaughan, 1974] and [Montgomery and Vaughan, 1973] (obtained at the same time by A. Selberg) is as follows.

Theorem (1974)

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

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

Theorem (1974)

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

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

The weighted large sieve inequality from Theorem 1 in [Montgomery and Vaughan, 1974] reads as follows.

Theorem (1974)

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

It is expected that one can reduce the constant \(c=3/2\) to 1. In this direction, we find in [Preissmann, 1984] the next result.

Theorem (1984)

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

See [Yangjit, 2023] for a discussion on this topic.