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

Abstract: The following generalisation of the Erd\H{o}s unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given $k$ positive real numbers $\delta_1,\dots,\delta_k$, a $(k+1)$-tuple $(p_1,\dots,p_{k+1})$ in $\mathbb{R}<sup>d$ is called a $(\delta,k)$-chain if $|p_j-p_{j+1}| = \delta_j$ for every $1\leq j \leq k$. What is the maximum number $C_k<sup>d(n)$ of $(k,\delta)$-chains in a set of $n$ points in $\mathbb{R}<sup>d$, where the maximum is taken over all $\delta$? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all $k$ in the planar case. error term It is only for $k\equiv 1$ (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 $3$ dimension.