The van der Corput property for sums of two squares

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in \mathbb{Z}\}\mathrm{.}$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong Sárközy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $\mathrm{\mid }A\mathrm{\mid }{\ll }_{\epsilon }{N}^{7\mathrm{/}8+\epsilon }$ for every $\epsilon >0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $\mathrm{\mid }A\mathrm{\mid }\gg N^{1\mathrm{/}2}$ such that $(A-A)\cap S_N=\mathrm{\varnothing }$.