Simplices in thin subsets of Euclidean spaces

TL;DR

This paper proves that in Euclidean spaces, sets with sufficiently large Hausdorff dimension necessarily contain similar copies of any given non-degenerate simplex, establishing a threshold dimension below which such copies may not exist.

Contribution

The authors identify a specific Hausdorff dimension threshold for sets in Euclidean spaces to contain similar simplices, advancing geometric measure theory.

Findings

Existence of a dimension threshold s_k<k for simplex containment

Sets with Hausdorff dimension at least s_k contain similar simplices

The threshold s_k is strictly less than the ambient space dimension k

Abstract

Let $\De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $A\subs \R^k$ of Hausdorff dimension $dim\,A\geq s_k$ necessarily contains a similar copy of the simplex $\De$.