Dot product chains

Abstract

We study a variant of Erd\H os' unit distance problem, concerning dot products between successive pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of nonzero dot products $({\alpha }_1,\dots ,{\alpha }_k)$, we give upper and lower bounds on the maximum possible number of tuples of distinct points $(A_1,\dots ,A_{k+1})\in E^{k+1}$ satisfying $A_j\cdot A_{j+1}={\alpha }_j$ for every $1\le j\le k$.

arXiv comment18 pages, 1 figure