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

Abstract

The following generalisation of the Erdős 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 ${\mathbb{R}}^d$ is called a $(\delta ,k)$-chain if $\Vert p_j-p_{j+1}\Vert ={\delta }_j$ for every $1\le j\le k$. What is the maximum number $C_k^d(n)$ of $(k,\delta )$-chains in a set of $n$ points in ${\mathbb{R}}^d$, 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.

New constructions and concluding remarks added