Bisectors and pinned distances

TL;DR

This paper establishes bounds on the number of pinned distances and bisector energy for small subsets in two-dimensional vector spaces over fields and the Euclidean plane, advancing understanding of geometric configurations.

Contribution

It provides new lower bounds on pinned distances over fields and upper bounds on bisector energy for finite Euclidean subsets, extending geometric combinatorics results.

Findings

Lower bounds on pinned distances in vector spaces over fields

Upper bounds on bisector energy in Euclidean plane

Advances in geometric combinatorics understanding

Abstract

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their bisector energy.