Almost sharp bounds on the number of discrete chains in the plane

The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. For a sequence δ = (δ1,..., δk) of k distances, a (k + 1)-tuple (p1,..., pk+1) of distinct points in Rd is called a (k, δ)-chain if kpj − pj+1k = δj for every 1 ≤ j ≤ k. What is the maximum number Ckd(n) of (k, δ)-chains in a set of n points in Rd, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. It is only for k ≡ 1 (mod 3) that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in dimension 3.