On distinct perpendicular bisectors and pinned distances in finite fields

Abstract

Given a set of points $P\subset {\mathbb{F}}_q^2$ such that $|P|\ge q^{3/2}$ it is established that $|P|$ determines $\Omega (q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P|\ge q^{4/3}$, then for a positive proportion of points $a\in P$, we have $$|\{\Vert a-b\Vert :b\in P\}|=\Omega (q),$$ where $\Vert a-b\Vert$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).

Cited by in corpus (4)

• An Improved Point-Line Incidence Bound Over Arbitrary Fields
• On the Pinned Distances Problem in Positive Characteristic
• Bisectors and pinned distances
• Almost spanning distance trees in subsets of finite vector spaces