Configurations in Fractals

Abstract

We define the manifold of configurations to be the quotient set of $k$ points in Euclidean space identified under congruence, and prove that compact subsets of $\mathbb{R}^d, d \geq 2$, of large Hausdorff dimension have a non-null set of configurations in them. Our method simplifies previous work in arXiv:1708.05919 and achieves a better dimensional threshold.