On the Number of Discrete Chains

We study a generalization of the Erdős unit distance problem to chains of k k distances. Given P , \mathcal {P}, a set of n n points, and a sequence of distances ( δ 1 , … , δ k ) (\delta _1,\ldots ,\delta _k) , we study the maximum possible number of tuples of distinct points ( p 1 , … , p k + 1 ) ∈ P k + 1 (p_1,\ldots ,p_{k+1})\in \mathcal {P}^{k+1} satisfying | p j p j + 1 | = δ j |p_jp_{j+1}|=\delta _j for every 1 ≤ j ≤ k 1\le j \le k . We study the problem in R 2 \mathbb {R}^2 and in R 3 \mathbb {R}^3 , and derive upper and lower bounds for this family of problems.