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arXiv · 2606.29185

The van der Corput property for sums of two squares

Steve Fan·Andrew Lott
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Abstract

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong S\'{a}rk\"{o}zy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon>0$, 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 $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.

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Steve Fan, Andrew Lott. 2026-06-28. The van der Corput property for sums of two squares. https://arxiv.org/abs/2606.29185

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