On the inner product problem in the plane

TL;DR

This paper proves the optimal lower bound of approximately N for the number of distinct inner products among N vectors in the plane by connecting incidence geometry and planar graph structures.

Contribution

It introduces a novel approach by lifting the problem to three dimensions and analyzing the resulting planar graphs to establish the bound.

Findings

Established the optimal lower bound of ~N for distinct inner products in the plane.

Connected incidence structures to planar graphs to derive bounds.

Provided a new geometric approach to inner product problems.

Abstract

We establish the optimal lower bound $\gtrsim N$ for counting the number of distinct inner products of pairs from any $N$ given vectors in $\R^2$. Essentially, we lift a related incidence structure defined by inner products in the plane to $\R^3$ and derive our bound by showing that the lifted structures result in planar graphs.