Simplices in thin subsets of Euclidean spaces

Iosevich, Alex; Magyar, Ákos

doi:10.2140/apde.2023.16.1485

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Simplices in thin subsets of Euclidean spaces

Alex Iosevich and Ákos Magyar

Vol. 16 (2023), No. 7, 1485–1496

DOI: 10.2140/apde.2023.16.1485

Abstract

Let $Δ$ be a nondegenerate simplex on $k$ vertices. We prove that there exists a threshold $s_{k} < k$ such that any set $A \subseteq ℝ^{k}$ of Hausdorff dimension $\dim A \geq s_{k}$ necessarily contains a similar copy of the simplex $Δ$ .

Keywords

simplexes, Hausdorff dimension, graphs

Mathematical Subject Classification

Primary: 28A75, 42B15

Milestones

Received: 7 September 2020

Revised: 28 September 2021

Accepted: 19 November 2021

Published: 21 September 2023

Authors

Alex Iosevich
Department of Mathematics University of Rochester Rochester, NY United States

Ákos Magyar
Department of Mathematics The University of Georgia Athens, GA United States

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Vol. 16, No. 7, 2023