On necklaces inside thin subsets of ${\mathbb{R}}^d$

We study similarity classes of point configurations in ${\mathbb{R}}^d$. Given a finite collection of points, a well-known question is: How high does the Hausdorff dimension $\text{hd}(E)$ of a compact set $E\subset {\mathbb{R}}^d$, $d\ge 2$, need to be to ensure that $E$ contains some similar copy of this configuration? We prove results for a related problem, showing that for $\text{hd}(D)$ sufficiently large, $E$ must contain many point configurations that we call $k$-necklaces of constant gap, generalizing equilateral triangles and rhombuses in higher dimensions. Our results extend and complement those in \cite{CLP14,BIT14}, where related questions were recently studied.