The optimal van der Corput property for squares

Abstract

We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large $N$, we construct a real trigonometric polynomial $P(\theta )=a_0+{\sum }_{1\le n\le \sqrt{N}}{a}_n\cos (2\pi n^2\theta )$ satisfying \begin{equation}\tag{1}\label{eq:abstract-vdc} P(θ)\ge0\quad(θ\in\mathbb{R}/\mathbb{Z}),\qquad P(0)=1, \end{equation} with \[0\le a_0\le\exp(-c\sqrt{\log N\log\log N}),\] where $c>0$ is an absolute constant. As a consequence, if $A\subseteq \mathbb{N}\cap [1,N]$ has no nonzero square difference, then \[|A|\le N\exp(-c\sqrt{\log N\log\log N}),\] improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević--Ninčević, and Fan--Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \eqref{eq:abstract-vdc} has \[a_0\ge\exp(-C\sqrt{\log N\log\log N})\] for an absolute constant $C>0$. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg--Sárközy theorem.

arXiv comment26 pages. Included more motivation for the proofs. Changed title and abstract to emphasize the sharp van der Corput bound