Congruent copies of finite patterns in planar point sets

Given a finite nonempty planar point set $S$, what is the maximum number of congruent copies of $S$ contained in a set of $n$ points in the Euclidean plane? Building on OpenAI’s recent breakthrough on the unit distance problem, we construct planar sets consisting of $n$ points that contain ${\mathrm{\Omega }}_S(n^{1+{\delta }_S})$ congruent copies of $S$, for some positive constant ${\delta }_S$ depending only on $S$. This answers a question of Brass and Pach in a strong form, and makes progress on questions posed by Erdős and Purdy, and Ábrego and Fernández-Merchant. Our proof uses the number field construction from Sawin’s quantitative refinement of OpenAI’s result and consequently yields an explicit choice for ${\delta }_S$ for each fixed $S$.