An optimal version of Sarkozy's theorem

Using Fourier analytic techniques, we prove that if $\VE>0$, $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and $A\subseteq\{1,...,N\}$, then there must exist $t\in\N$ such that $$\frac{|A\cap (A+t^2)|}{N}>(\frac{|A|}{N})^2-\VE.$$ This is a special case of results presented in Lyall and Magyar \cite{LM3} and we will follow those arguments closely. We hope that the exposition of this special case will serve to illuminate the key ideas contained in \cite{LM3}, where many of the analogous arguments are significantly more technical.