  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.
