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

We study similarity classes of point configurations in $\R^d$. Given a finite collection of points, a well-known question is: How high does the Hausdorff dimension $\hd(E)$ of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, need to be to ensure that $E$ contains some similar copy of this configuration? We prove results for a related problem, showing that for $\hd(D)$ 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.