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A Mattila-Sjölin theorem for simplices in low dimensions
Eyvindur Ari Palsson1, Francisco Romero Acosta1
1Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 USA.
Mathematische Annalen
|January 8, 2025
Summary
This study shows that compact sets with a sufficiently large Hausdorff dimension guarantee a non-empty interior for congruence classes of simplices. This finding improves existing results for triangles and extends to all simplices.
Area of Science:
- Geometric measure theory
- Fractal geometry
- Harmonic analysis
Background:
- The study of geometric properties of sets, particularly their dimensions, is crucial in understanding their structure.
- Previous work, like the Mattila-Sjölin theorem, established conditions for non-empty interiors of distance sets.
Purpose of the Study:
- To determine the minimum Hausdorff dimension of a compact set required to ensure a non-empty interior for the set of congruence classes of simplices formed by its points.
- To extend and improve upon existing theorems regarding the geometric properties of point sets and their associated congruence classes.
Main Methods:
- Utilizing concepts from fractal geometry and geometric measure theory.
- Analyzing the properties of congruence classes of simplices within a compact set E.
- Establishing a threshold for the Hausdorff dimension of E.
Main Results:
- A compact set E with Hausdorff dimension greater than d/2 (for d=2) or d-1 (for d>=3) guarantees a non-empty interior for its set of congruence classes of simplices.
- This result improves upon prior findings, particularly for the case of triangles (d=2).
- The findings are extended to all types of simplices, not just triangles.
Conclusions:
- The Hausdorff dimension plays a critical role in determining the topological properties of congruence classes of simplices generated by points in a compact set.
- This work advances the understanding of geometric structures and their congruence classes, building upon established theorems in the field.
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