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arXiv · 2608.19253

Simplex--center configurations in dense subsets of Euclidean spaces and the integer lattice

Akos Magyar
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Abstract

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.

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Akos Magyar. 2026-08-16. Simplex--center configurations in dense subsets of Euclidean spaces and the integer lattice. https://arxiv.org/abs/2608.19253

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