On the inner product problem in the plane

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 ${\mathbb{R}}^2$. Essentially, we lift a related incidence structure defined by inner products in the plane to ${\mathbb{R}}^3$ and derive our bound by showing that the lifted structures result in planar graphs.

Mistakes and gaps in the main construction