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 ${\mathbb{R}}^d,d\ge 2$, 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.