Angle chains and pinned variants

We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles $({\alpha }_1,\dots ,{\alpha }_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(x_1,\dots, x_{k+2})\in E^{k+2}$satisfying$\angle (x_j,x_{j+1},x_{j+2})=\alpha_j$forevery$1\le j \le k$ as well as pinned analogues.