Congruent copies of finite patterns in planar point sets

Shubhrajit Bhattacharya; Ritesh Goenka

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arXiv · 2606.27352

Congruent copies of finite patterns in planar point sets

Shubhrajit Bhattacharya·Ritesh Goenka

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Abstract

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 $\Omega_S(n^{1+\delta_S})$ 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 Erd\H{o}s and Purdy, and \'Abrego and Fern\'andez-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 $S$.

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Shubhrajit Bhattacharya, Ritesh Goenka. 2026-06-25. Congruent copies of finite patterns in planar point sets. https://arxiv.org/abs/2606.27352

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