On the inner product problem in the plane

Zhipeng Lu

On the inner product problem in the plane

Combinatorics 2022-01-13 v4 Metric Geometry Number Theory

Authors: Zhipeng Lu

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.

Keywords

incidence geometry of points and lines upper bounds optimization and approximation theory

Cite

@article{arxiv.2107.06807,
  title  = {On the inner product problem in the plane},
  author = {Zhipeng Lu},
  journal= {arXiv preprint arXiv:2107.06807},
  year   = {2022}
}

Comments

Mistakes and gaps in the main construction

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