The large sieve with quadratic amplitude

Abstract

We establish a large sieve bound for expressions of the form $$\sum\limits_{r=1}^R \left\vert \sum\limits_{M < n\le M+N} a_ne\left(\alpha_rf(n)\right)\right\vert^2,$$ where $f(x)=\alpha x^2+\beta x+\theta\in \mathbb{R}[x]$ is a quadratic polynomial with $\alpha>0$ and $\beta\ge 0$. We also consider the case when $f(x)=x^d$ with $d\in \mathbb{N}$, $d\ge 3$.