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

TL;DR

This paper determines the maximum number of specific discrete chains with fixed distances in planar point sets, improving previous bounds and providing near-optimal results for certain cases in three dimensions.

Contribution

It offers almost sharp bounds on the number of discrete chains in the plane for all lengths, extending and refining prior results on distance configurations.

Findings

Exact bounds for chains in the plane for all lengths.

Almost sharp bounds for chains in three dimensions for even lengths.

Dependence of results on maximum unit distances for certain chain lengths.

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 $\mathbb{R}^d$ 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 $C_k^d(n)$ 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.