A new proof of Sarkozy's theorem

It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect square. In this article we present a new and simple proof of this result by adapting an argument originally developed by Croot and Sisask to give a new proof of Roth's theorem.Comment: to appear in the Proc. Amer. Math. So

Optimal Polynomial Recurrence

Let $P\in\Z[n]$ with $P(0)=0$ and \VE>0. We show, using Fourier analytic techniques, that if N\geq \exp\exp(C\VE^{-1}\log\VE^{-1}) and $A\subseteq\{1,\...,N\}$, then there must exist $n\in\N$ such that \frac{|A\cap (A+P(n))|}{N}>(\frac{|A|}{N})^2-\VE. In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\N$, then the set of \emph{\VE-optimal return times} R(A,P,\VE)=\{n\in \N \,:\,\D(A\cap(A+P(n)))>\D(A)^2-\VE\} is syndetic for every \VE>0. Moreover, we show that R(A,P,\VE) is \emph{dense} in every sufficiently long interval, in the sense that there exists an L=L(\VE,P,A) such that |R(A,P,\VE)\cap I| \geq c(\VE,P)|I| for all intervals $I$ of natural numbers with $|I|\geq L$ and c(\VE,P)=\exp\exp(-C\,\VE^{-1}\log\VE^{-1}).Comment: Short remark added and typos fixe