On the Number of Discrete Chains

Abstract

We study a generalization of Erd\H os's unit distances problem to chains of $k$ distances. Given $\mathcal{P},$ a set of $n$ points, and a sequence of distances $({\delta }_1,\dots ,{\delta }_k)$, we study the maximum possible number of tuples of distinct points $(p_1,\dots ,p_{k+1})\in {\mathcal{P}}^{k+1}$ satisfying $|p_j{p}_{j+1}|={\delta }_j$ for every $1\le j\le k$. We study the problem in ${\mathbb{R}}^2$ and in ${\mathbb{R}}^3$, and derive upper and lower bounds for this family of problems.

9 pages, 1 figure