A Mattila–Sjölin theorem for simplices in low dimensions
2025
A Mattila–Sjölin theorem for simplices in low dimensions
publication
Evidence: high
Author Information
Author(s): Palsson Eyvindur Ari, Romero Acosta Francisco
Primary Institution: Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
Hypothesis
What is the Hausdorff dimension threshold needed to guarantee that the set of congruence classes of simplices has nonempty interior?
Conclusion
The study establishes a Hausdorff dimension threshold that guarantees the nonempty interior of congruence classes of simplices in low dimensions.
Supporting Evidence
- The study improves previous results by providing a new Hausdorff dimension threshold.
- It extends the Mattila–Sjölin theorem to include simplices beyond triangles.
- The results are applicable for dimensions greater than or equal to three.
Takeaway
If you have a compact set of points, and if the set is big enough, you can find shapes like triangles or higher-dimensional simplices that fit inside it without any gaps.
Methodology
The paper uses mathematical proofs to establish thresholds for Hausdorff dimensions related to simplices.
Digital Object Identifier (DOI)
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